Article ZFMISC_1, MML version 4.99.1005

:: ZFMISC_1:funcreg 1
registration
  let a1, a2 be set;
  cluster [a1,a2] -> non empty;
end;

:: ZFMISC_1:funcnot 1 => ZFMISC_1:func 1
definition
  let a1 be set;
  func bool A1 -> set means
    for b1 being set holds
          b1 in it
       iff
          b1 c= a1;
end;

:: ZFMISC_1:def 1
theorem
for b1, b2 being set holds
   b2 = bool b1
iff
   for b3 being set holds
         b3 in b2
      iff
         b3 c= b1;

:: ZFMISC_1:funcnot 2 => ZFMISC_1:func 2
definition
  let a1, a2 be set;
  func [:A1,A2:] -> set means
    for b1 being set holds
          b1 in it
       iff
          ex b2, b3 being set st
             b2 in a1 & b3 in a2 & b1 = [b2,b3];
end;

:: ZFMISC_1:def 2
theorem
for b1, b2, b3 being set holds
   b3 = [:b1,b2:]
iff
   for b4 being set holds
         b4 in b3
      iff
         ex b5, b6 being set st
            b5 in b1 & b6 in b2 & b4 = [b5,b6];

:: ZFMISC_1:funcnot 3 => ZFMISC_1:func 3
definition
  let a1, a2, a3 be set;
  func [:A1,A2,A3:] -> set equals
    [:[:a1,a2:],a3:];
end;

:: ZFMISC_1:def 3
theorem
for b1, b2, b3 being set holds
[:b1,b2,b3:] = [:[:b1,b2:],b3:];

:: ZFMISC_1:funcnot 4 => ZFMISC_1:func 4
definition
  let a1, a2, a3, a4 be set;
  func [:A1,A2,A3,A4:] -> set equals
    [:[:a1,a2,a3:],a4:];
end;

:: ZFMISC_1:def 4
theorem
for b1, b2, b3, b4 being set holds
[:b1,b2,b3,b4:] = [:[:b1,b2,b3:],b4:];

:: ZFMISC_1:th 1
theorem
bool {} = {{}};

:: ZFMISC_1:th 2
theorem
union {} = {};

:: ZFMISC_1:th 6
theorem
for b1, b2 being set
      st {b1} c= {b2}
   holds b1 = b2;

:: ZFMISC_1:th 8
theorem
for b1, b2, b3 being set
      st {b1} = {b2,b3}
   holds b1 = b2;

:: ZFMISC_1:th 9
theorem
for b1, b2, b3 being set
      st {b1} = {b2,b3}
   holds b2 = b3;

:: ZFMISC_1:th 10
theorem
for b1, b2, b3, b4 being set
      st {b1,b2} = {b3,b4} & b1 <> b3
   holds b1 = b4;

:: ZFMISC_1:th 12
theorem
for b1, b2 being set holds
{b1} c= {b1,b2};

:: ZFMISC_1:th 13
theorem
for b1, b2 being set
      st {b1} \/ {b2} = {b1}
   holds b1 = b2;

:: ZFMISC_1:th 14
theorem
for b1, b2 being set holds
{b1} \/ {b1,b2} = {b1,b2};

:: ZFMISC_1:th 16
theorem
for b1, b2 being set
      st {b1} misses {b2}
   holds b1 <> b2;

:: ZFMISC_1:th 17
theorem
for b1, b2 being set
      st b1 <> b2
   holds {b1} misses {b2};

:: ZFMISC_1:th 18
theorem
for b1, b2 being set
      st {b1} /\ {b2} = {b1}
   holds b1 = b2;

:: ZFMISC_1:th 19
theorem
for b1, b2 being set holds
{b1} /\ {b1,b2} = {b1};

:: ZFMISC_1:th 20
theorem
for b1, b2 being set holds
   {b1} \ {b2} = {b1}
iff
   b1 <> b2;

:: ZFMISC_1:th 21
theorem
for b1, b2 being set
      st {b1} \ {b2} = {}
   holds b1 = b2;

:: ZFMISC_1:th 22
theorem
for b1, b2 being set holds
{b1} \ {b1,b2} = {};

:: ZFMISC_1:th 23
theorem
for b1, b2 being set
      st b1 <> b2
   holds {b1,b2} \ {b2} = {b1};

:: ZFMISC_1:th 24
theorem
for b1, b2 being set
      st {b1} c= {b2}
   holds b1 = b2;

:: ZFMISC_1:th 25
theorem
for b1, b2, b3 being set
      st {b1} c= {b2,b3} & b1 <> b2
   holds b1 = b3;

:: ZFMISC_1:th 26
theorem
for b1, b2, b3 being set
      st {b1,b2} c= {b3}
   holds b1 = b3;

:: ZFMISC_1:th 27
theorem
for b1, b2, b3 being set
      st {b1,b2} c= {b3}
   holds {b1,b2} = {b3};

:: ZFMISC_1:th 28
theorem
for b1, b2, b3, b4 being set
      st {b1,b2} c= {b3,b4} & b1 <> b3
   holds b1 = b4;

:: ZFMISC_1:th 29
theorem
for b1, b2 being set
      st b1 <> b2
   holds {b1} \+\ {b2} = {b1,b2};

:: ZFMISC_1:th 30
theorem
for b1 being set holds
   bool {b1} = {{},{b1}};

:: ZFMISC_1:th 31
theorem
for b1 being set holds
   union {b1} = b1;

:: ZFMISC_1:th 32
theorem
for b1, b2 being set holds
union {{b1},{b2}} = {b1,b2};

:: ZFMISC_1:th 33
theorem
for b1, b2, b3, b4 being set
      st [b1,b2] = [b3,b4]
   holds b1 = b3 & b2 = b4;

:: ZFMISC_1:th 34
theorem
for b1, b2, b3, b4 being set holds
   [b1,b2] in [:{b3},{b4}:]
iff
   b1 = b3 & b2 = b4;

:: ZFMISC_1:th 35
theorem
for b1, b2 being set holds
[:{b1},{b2}:] = {[b1,b2]};

:: ZFMISC_1:th 36
theorem
for b1, b2, b3 being set holds
[:{b1},{b2,b3}:] = {[b1,b2],[b1,b3]} &
 [:{b1,b2},{b3}:] = {[b1,b3],[b2,b3]};

:: ZFMISC_1:th 37
theorem
for b1, b2 being set holds
   {b1} c= b2
iff
   b1 in b2;

:: ZFMISC_1:th 38
theorem
for b1, b2, b3 being set holds
   {b1,b2} c= b3
iff
   b1 in b3 & b2 in b3;

:: ZFMISC_1:th 39
theorem
for b1, b2 being set holds
   b1 c= {b2}
iff
   (b1 = {} or b1 = {b2});

:: ZFMISC_1:th 40
theorem
for b1, b2, b3 being set
      st b1 c= b2 & not b3 in b1
   holds b1 c= b2 \ {b3};

:: ZFMISC_1:th 41
theorem
for b1, b2 being set
      st b1 <> {b2} & b1 <> {}
   holds ex b3 being set st
      b3 in b1 & b3 <> b2;

:: ZFMISC_1:th 42
theorem
for b1, b2, b3 being set holds
   b1 c= {b2,b3}
iff
   (b1 <> {} & b1 <> {b2} & b1 <> {b3} implies b1 = {b2,b3});

:: ZFMISC_1:th 43
theorem
for b1, b2, b3 being set
      st {b1} = b2 \/ b3 &
         (b2 = {b1} implies b3 <> {b1}) &
         (b2 = {} implies b3 <> {b1})
   holds b2 = {b1} & b3 = {};

:: ZFMISC_1:th 44
theorem
for b1, b2, b3 being set
      st {b1} = b2 \/ b3 & b2 <> b3 & b2 <> {}
   holds b3 = {};

:: ZFMISC_1:th 45
theorem
for b1, b2 being set
      st {b1} \/ b2 c= b2
   holds b1 in b2;

:: ZFMISC_1:th 46
theorem
for b1, b2 being set
      st b1 in b2
   holds {b1} \/ b2 = b2;

:: ZFMISC_1:th 47
theorem
for b1, b2, b3 being set
      st {b1,b2} \/ b3 c= b3
   holds b1 in b3;

:: ZFMISC_1:th 48
theorem
for b1, b2, b3 being set
      st b1 in b2 & b3 in b2
   holds {b1,b3} \/ b2 = b2;

:: ZFMISC_1:th 49
theorem
for b1, b2 being set holds
{b1} \/ b2 <> {};

:: ZFMISC_1:th 50
theorem
for b1, b2, b3 being set holds
{b1,b2} \/ b3 <> {};

:: ZFMISC_1:th 51
theorem
for b1, b2 being set
      st b1 /\ {b2} = {b2}
   holds b2 in b1;

:: ZFMISC_1:th 52
theorem
for b1, b2 being set
      st b1 in b2
   holds b2 /\ {b1} = {b1};

:: ZFMISC_1:th 53
theorem
for b1, b2, b3 being set
      st b1 in b2 & b3 in b2
   holds {b1,b3} /\ b2 = {b1,b3};

:: ZFMISC_1:th 54
theorem
for b1, b2 being set
      st {b1} misses b2
   holds not b1 in b2;

:: ZFMISC_1:th 55
theorem
for b1, b2, b3 being set
      st {b1,b2} misses b3
   holds not b1 in b3;

:: ZFMISC_1:th 56
theorem
for b1, b2 being set
      st not b1 in b2
   holds {b1} misses b2;

:: ZFMISC_1:th 57
theorem
for b1, b2, b3 being set
      st not b1 in b2 & not b3 in b2
   holds {b1,b3} misses b2;

:: ZFMISC_1:th 58
theorem
for b1, b2 being set
      st {b1} meets b2
   holds {b1} /\ b2 = {b1};

:: ZFMISC_1:th 59
theorem
for b1, b2, b3 being set
      st {b1,b2} /\ b3 = {b1} &
         b2 in b3
   holds b1 = b2;

:: ZFMISC_1:th 60
theorem
for b1, b2, b3 being set
      st b1 in b2 & (b3 in b2 implies b1 = b3)
   holds {b1,b3} /\ b2 = {b1};

:: ZFMISC_1:th 63
theorem
for b1, b2, b3 being set
      st {b1,b2} /\ b3 = {b1,b2}
   holds b1 in b3;

:: ZFMISC_1:th 64
theorem
for b1, b2, b3 being set holds
   b1 in b2 \ {b3}
iff
   b1 in b2 & b1 <> b3;

:: ZFMISC_1:th 65
theorem
for b1, b2 being set holds
   b1 \ {b2} = b1
iff
   not b2 in b1;

:: ZFMISC_1:th 66
theorem
for b1, b2 being set
      st b1 \ {b2} = {} & b1 <> {}
   holds b1 = {b2};

:: ZFMISC_1:th 67
theorem
for b1, b2 being set holds
   {b1} \ b2 = {b1}
iff
   not b1 in b2;

:: ZFMISC_1:th 68
theorem
for b1, b2 being set holds
   {b1} \ b2 = {}
iff
   b1 in b2;

:: ZFMISC_1:th 69
theorem
for b1, b2 being set
      st {b1} \ b2 <> {}
   holds {b1} \ b2 = {b1};

:: ZFMISC_1:th 70
theorem
for b1, b2, b3 being set holds
   {b1,b2} \ b3 = {b1}
iff
   not b1 in b3 & (b2 in b3 or b1 = b2);

:: ZFMISC_1:th 72
theorem
for b1, b2, b3 being set holds
   {b1,b2} \ b3 = {b1,b2}
iff
   not b1 in b3 & not b2 in b3;

:: ZFMISC_1:th 73
theorem
for b1, b2, b3 being set holds
   {b1,b2} \ b3 = {}
iff
   b1 in b3 & b2 in b3;

:: ZFMISC_1:th 74
theorem
for b1, b2, b3 being set
      st {b1,b2} \ b3 <> {} &
         {b1,b2} \ b3 <> {b1} &
         {b1,b2} \ b3 <> {b2}
   holds {b1,b2} \ b3 = {b1,b2};

:: ZFMISC_1:th 75
theorem
for b1, b2, b3 being set holds
   b1 \ {b2,b3} = {}
iff
   (b1 <> {} & b1 <> {b2} & b1 <> {b3} implies b1 = {b2,b3});

:: ZFMISC_1:th 79
theorem
for b1, b2 being set
      st b1 c= b2
   holds bool b1 c= bool b2;

:: ZFMISC_1:th 80
theorem
for b1 being set holds
   {b1} c= bool b1;

:: ZFMISC_1:th 81
theorem
for b1, b2 being set holds
(bool b1) \/ bool b2 c= bool (b1 \/ b2);

:: ZFMISC_1:th 82
theorem
for b1, b2 being set
      st (bool b1) \/ bool b2 = bool (b1 \/ b2)
   holds b1,b2 are_c=-comparable;

:: ZFMISC_1:th 83
theorem
for b1, b2 being set holds
bool (b1 /\ b2) = (bool b1) /\ bool b2;

:: ZFMISC_1:th 84
theorem
for b1, b2 being set holds
bool (b1 \ b2) c= {{}} \/ ((bool b1) \ bool b2);

:: ZFMISC_1:th 86
theorem
for b1, b2 being set holds
(bool (b1 \ b2)) \/ bool (b2 \ b1) c= bool (b1 \+\ b2);

:: ZFMISC_1:th 92
theorem
for b1, b2 being set
      st b1 in b2
   holds b1 c= union b2;

:: ZFMISC_1:th 93
theorem
for b1, b2 being set holds
union {b1,b2} = b1 \/ b2;

:: ZFMISC_1:th 94
theorem
for b1, b2 being set
      st for b3 being set
              st b3 in b1
           holds b3 c= b2
   holds union b1 c= b2;

:: ZFMISC_1:th 95
theorem
for b1, b2 being set
      st b1 c= b2
   holds union b1 c= union b2;

:: ZFMISC_1:th 96
theorem
for b1, b2 being set holds
union (b1 \/ b2) = (union b1) \/ union b2;

:: ZFMISC_1:th 97
theorem
for b1, b2 being set holds
union (b1 /\ b2) c= (union b1) /\ union b2;

:: ZFMISC_1:th 98
theorem
for b1, b2 being set
      st for b3 being set
              st b3 in b1
           holds b3 misses b2
   holds union b1 misses b2;

:: ZFMISC_1:th 99
theorem
for b1 being set holds
   union bool b1 = b1;

:: ZFMISC_1:th 100
theorem
for b1 being set holds
   b1 c= bool union b1;

:: ZFMISC_1:th 101
theorem
for b1, b2 being set
      st for b3, b4 being set
              st b3 <> b4 & b3 in b1 \/ b2 & b4 in b1 \/ b2
           holds b3 misses b4
   holds union (b1 /\ b2) = (union b1) /\ union b2;

:: ZFMISC_1:th 103
theorem
for b1, b2, b3, b4 being set
      st b1 c= [:b2,b3:] & b4 in b1
   holds ex b5, b6 being set st
      b5 in b2 & b6 in b3 & b4 = [b5,b6];

:: ZFMISC_1:th 104
theorem
for b1, b2, b3, b4, b5 being set
      st b1 in [:b2,b3:] /\ [:b4,b5:]
   holds ex b6, b7 being set st
      b1 = [b6,b7] & b6 in b2 /\ b4 & b7 in b3 /\ b5;

:: ZFMISC_1:th 105
theorem
for b1, b2 being set holds
[:b1,b2:] c= bool bool (b1 \/ b2);

:: ZFMISC_1:th 106
theorem
for b1, b2, b3, b4 being set holds
   [b1,b2] in [:b3,b4:]
iff
   b1 in b3 & b2 in b4;

:: ZFMISC_1:th 107
theorem
for b1, b2, b3, b4 being set
      st [b1,b2] in [:b3,b4:]
   holds [b2,b1] in [:b4,b3:];

:: ZFMISC_1:th 108
theorem
for b1, b2, b3, b4 being set
      st for b5, b6 being set holds
           [b5,b6] in [:b1,b2:]
        iff
           [b5,b6] in [:b3,b4:]
   holds [:b1,b2:] = [:b3,b4:];

:: ZFMISC_1:th 113
theorem
for b1, b2 being set holds
   [:b1,b2:] = {}
iff
   (b1 = {} or b2 = {});

:: ZFMISC_1:th 114
theorem
for b1, b2 being set
      st b1 <> {} & b2 <> {} & [:b1,b2:] = [:b2,b1:]
   holds b1 = b2;

:: ZFMISC_1:th 115
theorem
for b1, b2 being set
      st [:b1,b1:] = [:b2,b2:]
   holds b1 = b2;

:: ZFMISC_1:th 116
theorem
for b1 being set
      st b1 c= [:b1,b1:]
   holds b1 = {};

:: ZFMISC_1:th 117
theorem
for b1, b2, b3 being set
      st b1 <> {} &
         ([:b2,b1:] c= [:b3,b1:] or [:b1,b2:] c= [:b1,b3:])
   holds b2 c= b3;

:: ZFMISC_1:th 118
theorem
for b1, b2, b3 being set
      st b1 c= b2
   holds [:b1,b3:] c= [:b2,b3:] & [:b3,b1:] c= [:b3,b2:];

:: ZFMISC_1:th 119
theorem
for b1, b2, b3, b4 being set
      st b1 c= b2 & b3 c= b4
   holds [:b1,b3:] c= [:b2,b4:];

:: ZFMISC_1:th 120
theorem
for b1, b2, b3 being set holds
[:b1 \/ b2,b3:] = [:b1,b3:] \/ [:b2,b3:] &
 [:b3,b1 \/ b2:] = [:b3,b1:] \/ [:b3,b2:];

:: ZFMISC_1:th 121
theorem
for b1, b2, b3, b4 being set holds
[:b1 \/ b2,b3 \/ b4:] = (([:b1,b3:] \/ [:b1,b4:]) \/ [:b2,b3:]) \/ [:b2,b4:];

:: ZFMISC_1:th 122
theorem
for b1, b2, b3 being set holds
[:b1 /\ b2,b3:] = [:b1,b3:] /\ [:b2,b3:] &
 [:b3,b1 /\ b2:] = [:b3,b1:] /\ [:b3,b2:];

:: ZFMISC_1:th 123
theorem
for b1, b2, b3, b4 being set holds
[:b1 /\ b2,b3 /\ b4:] = [:b1,b3:] /\ [:b2,b4:];

:: ZFMISC_1:th 124
theorem
for b1, b2, b3, b4 being set
      st b1 c= b2 & b3 c= b4
   holds [:b1,b4:] /\ [:b2,b3:] = [:b1,b3:];

:: ZFMISC_1:th 125
theorem
for b1, b2, b3 being set holds
[:b1 \ b2,b3:] = [:b1,b3:] \ [:b2,b3:] &
 [:b3,b1 \ b2:] = [:b3,b1:] \ [:b3,b2:];

:: ZFMISC_1:th 126
theorem
for b1, b2, b3, b4 being set holds
[:b1,b2:] \ [:b3,b4:] = [:b1 \ b3,b2:] \/ [:b1,b2 \ b4:];

:: ZFMISC_1:th 127
theorem
for b1, b2, b3, b4 being set
      st (b1 misses b2 or b3 misses b4)
   holds [:b1,b3:] misses [:b2,b4:];

:: ZFMISC_1:th 128
theorem
for b1, b2, b3, b4 being set holds
   [b1,b2] in [:{b3},b4:]
iff
   b1 = b3 & b2 in b4;

:: ZFMISC_1:th 129
theorem
for b1, b2, b3, b4 being set holds
   [b1,b2] in [:b3,{b4}:]
iff
   b1 in b3 & b2 = b4;

:: ZFMISC_1:th 130
theorem
for b1, b2 being set
      st b1 <> {}
   holds [:{b2},b1:] <> {} &
    [:b1,{b2}:] <> {};

:: ZFMISC_1:th 131
theorem
for b1, b2, b3, b4 being set
      st b1 <> b2
   holds [:{b1},b3:] misses [:{b2},b4:] &
    [:b3,{b1}:] misses [:b4,{b2}:];

:: ZFMISC_1:th 132
theorem
for b1, b2, b3 being set holds
[:{b1,b2},b3:] = [:{b1},b3:] \/ [:{b2},b3:] &
 [:b3,{b1,b2}:] = [:b3,{b1}:] \/ [:b3,{b2}:];

:: ZFMISC_1:th 134
theorem
for b1, b2, b3, b4 being set
      st b1 <> {} & b2 <> {} & [:b1,b2:] = [:b3,b4:]
   holds b1 = b3 & b2 = b4;

:: ZFMISC_1:th 135
theorem
for b1, b2 being set
      st (b1 c= [:b1,b2:] or b1 c= [:b2,b1:])
   holds b1 = {};

:: ZFMISC_1:th 136
theorem
for b1 being set holds
   ex b2 being set st
      b1 in b2 &
       (for b3, b4 being set
             st b3 in b2 & b4 c= b3
          holds b4 in b2) &
       (for b3 being set
             st b3 in b2
          holds bool b3 in b2) &
       (for b3 being set
             st b3 c= b2 & not b3,b2 are_equipotent
          holds b3 in b2);

:: ZFMISC_1:th 137
theorem
for b1, b2, b3, b4, b5 being set
      st b1 in [:b2,b3:] & b1 in [:b4,b5:]
   holds b1 in [:b2 /\ b4,b3 /\ b5:];

:: ZFMISC_1:th 138
theorem
for b1, b2, b3, b4 being set
      st [:b1,b2:] c= [:b3,b4:] & [:b1,b2:] <> {}
   holds b1 c= b3 & b2 c= b4;

:: ZFMISC_1:th 139
theorem
for b1 being non empty set
for b2, b3, b4 being set
      st ([:b1,b2:] c= [:b3,b4:] or [:b2,b1:] c= [:b4,b3:])
   holds b2 c= b4;

:: ZFMISC_1:th 140
theorem
for b1, b2 being set
      st b1 in b2
   holds (b2 \ {b1}) \/ {b1} = b2;

:: ZFMISC_1:th 141
theorem
for b1, b2 being set
      st not b1 in b2
   holds (b2 \/ {b1}) \ {b1} = b2;

:: ZFMISC_1:th 142
theorem
for b1, b2, b3, b4 being set holds
   b4 c= {b1,b2,b3}
iff
   (b4 <> {} & b4 <> {b1} & b4 <> {b2} & b4 <> {b3} & b4 <> {b1,b2} & b4 <> {b2,b3} & b4 <> {b1,b3} implies b4 = {b1,b2,b3});

:: ZFMISC_1:th 143
theorem
for b1, b2, b3, b4, b5, b6 being set
      st b1 c= [:b3,b4:] & b2 c= [:b5,b6:]
   holds b1 \/ b2 c= [:b3 \/ b5,b4 \/ b6:];

:: ZFMISC_1:th 144
theorem
for b1, b2, b3 being set
      st not b1 in b3 & not b2 in b3
   holds b3 = b3 \ {b1,b2};

:: ZFMISC_1:th 145
theorem
for b1, b2, b3 being set
      st not b1 in b3 & not b2 in b3
   holds b3 = (b3 \/ {b1,b2}) \ {b1,b2};

:: ZFMISC_1:prednot 1 => ZFMISC_1:pred 1
definition
  let a1, a2, a3 be set;
  pred A1,A2,A3 are_mutually_different means
    a1 <> a2 & a1 <> a3 & a2 <> a3;
end;

:: ZFMISC_1:dfs 5
definiens
  let a1, a2, a3 be set;
To prove
     a1,a2,a3 are_mutually_different
it is sufficient to prove
  thus a1 <> a2 & a1 <> a3 & a2 <> a3;

:: ZFMISC_1:def 5
theorem
for b1, b2, b3 being set holds
   b1,b2,b3 are_mutually_different
iff
   b1 <> b2 & b1 <> b3 & b2 <> b3;

:: ZFMISC_1:prednot 2 => ZFMISC_1:pred 2
definition
  let a1, a2, a3, a4 be set;
  pred A1,A2,A3,A4 are_mutually_different means
    a1 <> a2 & a1 <> a3 & a1 <> a4 & a2 <> a3 & a2 <> a4 & a3 <> a4;
end;

:: ZFMISC_1:dfs 6
definiens
  let a1, a2, a3, a4 be set;
To prove
     a1,a2,a3,a4 are_mutually_different
it is sufficient to prove
  thus a1 <> a2 & a1 <> a3 & a1 <> a4 & a2 <> a3 & a2 <> a4 & a3 <> a4;

:: ZFMISC_1:def 6
theorem
for b1, b2, b3, b4 being set holds
   b1,b2,b3,b4 are_mutually_different
iff
   b1 <> b2 & b1 <> b3 & b1 <> b4 & b2 <> b3 & b2 <> b4 & b3 <> b4;

:: ZFMISC_1:prednot 3 => ZFMISC_1:pred 3
definition
  let a1, a2, a3, a4, a5 be set;
  pred A1,A2,A3,A4,A5 are_mutually_different means
    a1 <> a2 & a1 <> a3 & a1 <> a4 & a1 <> a5 & a2 <> a3 & a2 <> a4 & a2 <> a5 & a3 <> a4 & a3 <> a5 & a4 <> a5;
end;

:: ZFMISC_1:dfs 7
definiens
  let a1, a2, a3, a4, a5 be set;
To prove
     a1,a2,a3,a4,a5 are_mutually_different
it is sufficient to prove
  thus a1 <> a2 & a1 <> a3 & a1 <> a4 & a1 <> a5 & a2 <> a3 & a2 <> a4 & a2 <> a5 & a3 <> a4 & a3 <> a5 & a4 <> a5;

:: ZFMISC_1:def 7
theorem
for b1, b2, b3, b4, b5 being set holds
   b1,b2,b3,b4,b5 are_mutually_different
iff
   b1 <> b2 & b1 <> b3 & b1 <> b4 & b1 <> b5 & b2 <> b3 & b2 <> b4 & b2 <> b5 & b3 <> b4 & b3 <> b5 & b4 <> b5;

:: ZFMISC_1:prednot 4 => ZFMISC_1:pred 4
definition
  let a1, a2, a3, a4, a5, a6 be set;
  pred A1,A2,A3,A4,A5,A6 are_mutually_different means
    a1 <> a2 & a1 <> a3 & a1 <> a4 & a1 <> a5 & a1 <> a6 & a2 <> a3 & a2 <> a4 & a2 <> a5 & a2 <> a6 & a3 <> a4 & a3 <> a5 & a3 <> a6 & a4 <> a5 & a4 <> a6 & a5 <> a6;
end;

:: ZFMISC_1:dfs 8
definiens
  let a1, a2, a3, a4, a5, a6 be set;
To prove
     a1,a2,a3,a4,a5,a6 are_mutually_different
it is sufficient to prove
  thus a1 <> a2 & a1 <> a3 & a1 <> a4 & a1 <> a5 & a1 <> a6 & a2 <> a3 & a2 <> a4 & a2 <> a5 & a2 <> a6 & a3 <> a4 & a3 <> a5 & a3 <> a6 & a4 <> a5 & a4 <> a6 & a5 <> a6;

:: ZFMISC_1:def 8
theorem
for b1, b2, b3, b4, b5, b6 being set holds
   b1,b2,b3,b4,b5,b6 are_mutually_different
iff
   b1 <> b2 & b1 <> b3 & b1 <> b4 & b1 <> b5 & b1 <> b6 & b2 <> b3 & b2 <> b4 & b2 <> b5 & b2 <> b6 & b3 <> b4 & b3 <> b5 & b3 <> b6 & b4 <> b5 & b4 <> b6 & b5 <> b6;

:: ZFMISC_1:prednot 5 => ZFMISC_1:pred 5
definition
  let a1, a2, a3, a4, a5, a6, a7 be set;
  pred A1,A2,A3,A4,A5,A6,A7 are_mutually_different means
    a1 <> a2 & a1 <> a3 & a1 <> a4 & a1 <> a5 & a1 <> a6 & a1 <> a7 & a2 <> a3 & a2 <> a4 & a2 <> a5 & a2 <> a6 & a2 <> a7 & a3 <> a4 & a3 <> a5 & a3 <> a6 & a3 <> a7 & a4 <> a5 & a4 <> a6 & a4 <> a7 & a5 <> a6 & a5 <> a7 & a6 <> a7;
end;

:: ZFMISC_1:dfs 9
definiens
  let a1, a2, a3, a4, a5, a6, a7 be set;
To prove
     a1,a2,a3,a4,a5,a6,a7 are_mutually_different
it is sufficient to prove
  thus a1 <> a2 & a1 <> a3 & a1 <> a4 & a1 <> a5 & a1 <> a6 & a1 <> a7 & a2 <> a3 & a2 <> a4 & a2 <> a5 & a2 <> a6 & a2 <> a7 & a3 <> a4 & a3 <> a5 & a3 <> a6 & a3 <> a7 & a4 <> a5 & a4 <> a6 & a4 <> a7 & a5 <> a6 & a5 <> a7 & a6 <> a7;

:: ZFMISC_1:def 9
theorem
for b1, b2, b3, b4, b5, b6, b7 being set holds
   b1,b2,b3,b4,b5,b6,b7 are_mutually_different
iff
   b1 <> b2 & b1 <> b3 & b1 <> b4 & b1 <> b5 & b1 <> b6 & b1 <> b7 & b2 <> b3 & b2 <> b4 & b2 <> b5 & b2 <> b6 & b2 <> b7 & b3 <> b4 & b3 <> b5 & b3 <> b6 & b3 <> b7 & b4 <> b5 & b4 <> b6 & b4 <> b7 & b5 <> b6 & b5 <> b7 & b6 <> b7;

:: ZFMISC_1:th 146
theorem
for b1, b2, b3, b4 being set holds
[:{b1,b2},{b3,b4}:] = {[b1,b3],[b1,b4],[b2,b3],[b2,b4]};