Article XBOOLE_1, MML version 4.99.1005

:: XBOOLE_1:th 1
theorem
for b1, b2, b3 being set
      st b1 c= b2 & b2 c= b3
   holds b1 c= b3;

:: XBOOLE_1:th 2
theorem
for b1 being set holds
   {} c= b1;

:: XBOOLE_1:th 3
theorem
for b1 being set
      st b1 c= {}
   holds b1 = {};

:: XBOOLE_1:th 4
theorem
for b1, b2, b3 being set holds
(b1 \/ b2) \/ b3 = b1 \/ (b2 \/ b3);

:: XBOOLE_1:th 5
theorem
for b1, b2, b3 being set holds
(b1 \/ b2) \/ b3 = (b1 \/ b3) \/ (b2 \/ b3);

:: XBOOLE_1:th 6
theorem
for b1, b2 being set holds
b1 \/ (b1 \/ b2) = b1 \/ b2;

:: XBOOLE_1:th 7
theorem
for b1, b2 being set holds
b1 c= b1 \/ b2;

:: XBOOLE_1:th 8
theorem
for b1, b2, b3 being set
      st b1 c= b2 & b3 c= b2
   holds b1 \/ b3 c= b2;

:: XBOOLE_1:th 9
theorem
for b1, b2, b3 being set
      st b1 c= b2
   holds b1 \/ b3 c= b2 \/ b3;

:: XBOOLE_1:th 10
theorem
for b1, b2, b3 being set
      st b1 c= b2
   holds b1 c= b3 \/ b2;

:: XBOOLE_1:th 11
theorem
for b1, b2, b3 being set
      st b1 \/ b2 c= b3
   holds b1 c= b3;

:: XBOOLE_1:th 12
theorem
for b1, b2 being set
      st b1 c= b2
   holds b1 \/ b2 = b2;

:: XBOOLE_1:th 13
theorem
for b1, b2, b3, b4 being set
      st b1 c= b2 & b3 c= b4
   holds b1 \/ b3 c= b2 \/ b4;

:: XBOOLE_1:th 14
theorem
for b1, b2, b3 being set
      st b1 c= b2 &
         b3 c= b2 &
         (for b4 being set
               st b1 c= b4 & b3 c= b4
            holds b2 c= b4)
   holds b2 = b1 \/ b3;

:: XBOOLE_1:th 15
theorem
for b1, b2 being set
      st b1 \/ b2 = {}
   holds b1 = {};

:: XBOOLE_1:th 16
theorem
for b1, b2, b3 being set holds
(b1 /\ b2) /\ b3 = b1 /\ (b2 /\ b3);

:: XBOOLE_1:th 17
theorem
for b1, b2 being set holds
b1 /\ b2 c= b1;

:: XBOOLE_1:th 18
theorem
for b1, b2, b3 being set
      st b1 c= b2 /\ b3
   holds b1 c= b2;

:: XBOOLE_1:th 19
theorem
for b1, b2, b3 being set
      st b1 c= b2 & b1 c= b3
   holds b1 c= b2 /\ b3;

:: XBOOLE_1:th 20
theorem
for b1, b2, b3 being set
      st b1 c= b2 &
         b1 c= b3 &
         (for b4 being set
               st b4 c= b2 & b4 c= b3
            holds b4 c= b1)
   holds b1 = b2 /\ b3;

:: XBOOLE_1:th 21
theorem
for b1, b2 being set holds
b1 /\ (b1 \/ b2) = b1;

:: XBOOLE_1:th 22
theorem
for b1, b2 being set holds
b1 \/ (b1 /\ b2) = b1;

:: XBOOLE_1:th 23
theorem
for b1, b2, b3 being set holds
b1 /\ (b2 \/ b3) = (b1 /\ b2) \/ (b1 /\ b3);

:: XBOOLE_1:th 24
theorem
for b1, b2, b3 being set holds
b1 \/ (b2 /\ b3) = (b1 \/ b2) /\ (b1 \/ b3);

:: XBOOLE_1:th 25
theorem
for b1, b2, b3 being set holds
((b1 /\ b2) \/ (b2 /\ b3)) \/ (b3 /\ b1) = ((b1 \/ b2) /\ (b2 \/ b3)) /\ (b3 \/ b1);

:: XBOOLE_1:th 26
theorem
for b1, b2, b3 being set
      st b1 c= b2
   holds b1 /\ b3 c= b2 /\ b3;

:: XBOOLE_1:th 27
theorem
for b1, b2, b3, b4 being set
      st b1 c= b2 & b3 c= b4
   holds b1 /\ b3 c= b2 /\ b4;

:: XBOOLE_1:th 28
theorem
for b1, b2 being set
      st b1 c= b2
   holds b1 /\ b2 = b1;

:: XBOOLE_1:th 29
theorem
for b1, b2, b3 being set holds
b1 /\ b2 c= b1 \/ b3;

:: XBOOLE_1:th 30
theorem
for b1, b2, b3 being set
      st b1 c= b2
   holds b1 \/ (b3 /\ b2) = (b1 \/ b3) /\ b2;

:: XBOOLE_1:th 31
theorem
for b1, b2, b3 being set holds
(b1 /\ b2) \/ (b1 /\ b3) c= b2 \/ b3;

:: XBOOLE_1:th 32
theorem
for b1, b2 being set
      st b1 \ b2 = b2 \ b1
   holds b1 = b2;

:: XBOOLE_1:th 33
theorem
for b1, b2, b3 being set
      st b1 c= b2
   holds b1 \ b3 c= b2 \ b3;

:: XBOOLE_1:th 34
theorem
for b1, b2, b3 being set
      st b1 c= b2
   holds b3 \ b2 c= b3 \ b1;

:: XBOOLE_1:th 35
theorem
for b1, b2, b3, b4 being set
      st b1 c= b2 & b3 c= b4
   holds b1 \ b4 c= b2 \ b3;

:: XBOOLE_1:th 36
theorem
for b1, b2 being set holds
b1 \ b2 c= b1;

:: XBOOLE_1:th 37
theorem
for b1, b2 being set holds
   b1 \ b2 = {}
iff
   b1 c= b2;

:: XBOOLE_1:th 38
theorem
for b1, b2 being set
      st b1 c= b2 \ b1
   holds b1 = {};

:: XBOOLE_1:th 39
theorem
for b1, b2 being set holds
b1 \/ (b2 \ b1) = b1 \/ b2;

:: XBOOLE_1:th 40
theorem
for b1, b2 being set holds
(b1 \/ b2) \ b2 = b1 \ b2;

:: XBOOLE_1:th 41
theorem
for b1, b2, b3 being set holds
(b1 \ b2) \ b3 = b1 \ (b2 \/ b3);

:: XBOOLE_1:th 42
theorem
for b1, b2, b3 being set holds
(b1 \/ b2) \ b3 = (b1 \ b3) \/ (b2 \ b3);

:: XBOOLE_1:th 43
theorem
for b1, b2, b3 being set
      st b1 c= b2 \/ b3
   holds b1 \ b2 c= b3;

:: XBOOLE_1:th 44
theorem
for b1, b2, b3 being set
      st b1 \ b2 c= b3
   holds b1 c= b2 \/ b3;

:: XBOOLE_1:th 45
theorem
for b1, b2 being set
      st b1 c= b2
   holds b2 = b1 \/ (b2 \ b1);

:: XBOOLE_1:th 46
theorem
for b1, b2 being set holds
b1 \ (b1 \/ b2) = {};

:: XBOOLE_1:th 47
theorem
for b1, b2 being set holds
b1 \ (b1 /\ b2) = b1 \ b2;

:: XBOOLE_1:th 48
theorem
for b1, b2 being set holds
b1 \ (b1 \ b2) = b1 /\ b2;

:: XBOOLE_1:th 49
theorem
for b1, b2, b3 being set holds
b1 /\ (b2 \ b3) = (b1 /\ b2) \ b3;

:: XBOOLE_1:th 50
theorem
for b1, b2, b3 being set holds
b1 /\ (b2 \ b3) = (b1 /\ b2) \ (b1 /\ b3);

:: XBOOLE_1:th 51
theorem
for b1, b2 being set holds
(b1 /\ b2) \/ (b1 \ b2) = b1;

:: XBOOLE_1:th 52
theorem
for b1, b2, b3 being set holds
b1 \ (b2 \ b3) = (b1 \ b2) \/ (b1 /\ b3);

:: XBOOLE_1:th 53
theorem
for b1, b2, b3 being set holds
b1 \ (b2 \/ b3) = (b1 \ b2) /\ (b1 \ b3);

:: XBOOLE_1:th 54
theorem
for b1, b2, b3 being set holds
b1 \ (b2 /\ b3) = (b1 \ b2) \/ (b1 \ b3);

:: XBOOLE_1:th 55
theorem
for b1, b2 being set holds
(b1 \/ b2) \ (b1 /\ b2) = (b1 \ b2) \/ (b2 \ b1);

:: XBOOLE_1:th 56
theorem
for b1, b2, b3 being set
      st b1 c< b2 & b2 c< b3
   holds b1 c< b3;

:: XBOOLE_1:th 57
theorem
for b1, b2 being set
      st b1 c< b2
   holds not b2 c< b1;

:: XBOOLE_1:th 58
theorem
for b1, b2, b3 being set
      st b1 c< b2 & b2 c= b3
   holds b1 c< b3;

:: XBOOLE_1:th 59
theorem
for b1, b2, b3 being set
      st b1 c= b2 & b2 c< b3
   holds b1 c< b3;

:: XBOOLE_1:th 60
theorem
for b1, b2 being set
      st b1 c= b2
   holds not b2 c< b1;

:: XBOOLE_1:th 61
theorem
for b1 being set
      st b1 <> {}
   holds {} c< b1;

:: XBOOLE_1:th 62
theorem
for b1 being set holds
   not b1 c< {};

:: XBOOLE_1:th 63
theorem
for b1, b2, b3 being set
      st b1 c= b2 & b2 misses b3
   holds b1 misses b3;

:: XBOOLE_1:th 64
theorem
for b1, b2, b3, b4 being set
      st b1 c= b2 & b3 c= b4 & b2 misses b4
   holds b1 misses b3;

:: XBOOLE_1:th 65
theorem
for b1 being set holds
   b1 misses {};

:: XBOOLE_1:th 66
theorem
for b1 being set holds
      b1 meets b1
   iff
      b1 <> {};

:: XBOOLE_1:th 67
theorem
for b1, b2, b3 being set
      st b1 c= b2 & b1 c= b3 & b2 misses b3
   holds b1 = {};

:: XBOOLE_1:th 68
theorem
for b1, b2 being set
for b3 being non empty set
      st b3 c= b1 & b3 c= b2
   holds b1 meets b2;

:: XBOOLE_1:th 69
theorem
for b1 being set
for b2 being non empty set
      st b2 c= b1
   holds b2 meets b1;

:: XBOOLE_1:th 70
theorem
for b1, b2, b3 being set holds
   b1 meets b2 \/ b3
iff
   (b1 misses b2 implies b1 meets b3);

:: XBOOLE_1:th 71
theorem
for b1, b2, b3 being set
      st b1 \/ b2 = b3 \/ b2 & b1 misses b2 & b3 misses b2
   holds b1 = b3;

:: XBOOLE_1:th 72
theorem
for b1, b2, b3, b4 being set
      st b1 \/ b2 = b3 \/ b4 & b3 misses b1 & b4 misses b2
   holds b3 = b2;

:: XBOOLE_1:th 73
theorem
for b1, b2, b3 being set
      st b1 c= b2 \/ b3 & b1 misses b3
   holds b1 c= b2;

:: XBOOLE_1:th 74
theorem
for b1, b2, b3 being set
      st b1 meets b2 /\ b3
   holds b1 meets b2;

:: XBOOLE_1:th 75
theorem
for b1, b2 being set
      st b1 meets b2
   holds b1 /\ b2 meets b2;

:: XBOOLE_1:th 76
theorem
for b1, b2, b3 being set
      st b1 misses b2
   holds b3 /\ b1 misses b3 /\ b2;

:: XBOOLE_1:th 77
theorem
for b1, b2, b3 being set
      st b1 meets b2 & b1 c= b3
   holds b1 meets b2 /\ b3;

:: XBOOLE_1:th 78
theorem
for b1, b2, b3 being set
      st b1 misses b2
   holds b1 /\ (b2 \/ b3) = b1 /\ b3;

:: XBOOLE_1:th 79
theorem
for b1, b2 being set holds
b1 \ b2 misses b2;

:: XBOOLE_1:th 80
theorem
for b1, b2, b3 being set
      st b1 misses b2
   holds b1 misses b2 \ b3;

:: XBOOLE_1:th 81
theorem
for b1, b2, b3 being set
      st b1 misses b2 \ b3
   holds b2 misses b1 \ b3;

:: XBOOLE_1:th 82
theorem
for b1, b2 being set holds
b1 \ b2 misses b2 \ b1;

:: XBOOLE_1:th 83
theorem
for b1, b2 being set holds
   b1 misses b2
iff
   b1 \ b2 = b1;

:: XBOOLE_1:th 84
theorem
for b1, b2, b3 being set
      st b1 meets b2 & b1 misses b3
   holds b1 meets b2 \ b3;

:: XBOOLE_1:th 85
theorem
for b1, b2, b3 being set
      st b1 c= b2
   holds b1 misses b3 \ b2;

:: XBOOLE_1:th 86
theorem
for b1, b2, b3 being set
      st b1 c= b2 & b1 misses b3
   holds b1 c= b2 \ b3;

:: XBOOLE_1:th 87
theorem
for b1, b2, b3 being set
      st b1 misses b2
   holds (b3 \ b1) \/ b2 = (b3 \/ b2) \ b1;

:: XBOOLE_1:th 88
theorem
for b1, b2 being set
      st b1 misses b2
   holds (b1 \/ b2) \ b2 = b1;

:: XBOOLE_1:th 89
theorem
for b1, b2 being set holds
b1 /\ b2 misses b1 \ b2;

:: XBOOLE_1:th 90
theorem
for b1, b2 being set holds
b1 \ (b1 /\ b2) misses b2;

:: XBOOLE_1:th 91
theorem
for b1, b2, b3 being set holds
(b1 \+\ b2) \+\ b3 = b1 \+\ (b2 \+\ b3);

:: XBOOLE_1:th 92
theorem
for b1 being set holds
   b1 \+\ b1 = {};

:: XBOOLE_1:th 93
theorem
for b1, b2 being set holds
b1 \/ b2 = (b1 \+\ b2) \/ (b1 /\ b2);

:: XBOOLE_1:th 94
theorem
for b1, b2 being set holds
b1 \/ b2 = (b1 \+\ b2) \+\ (b1 /\ b2);

:: XBOOLE_1:th 95
theorem
for b1, b2 being set holds
b1 /\ b2 = (b1 \+\ b2) \+\ (b1 \/ b2);

:: XBOOLE_1:th 96
theorem
for b1, b2 being set holds
b1 \ b2 c= b1 \+\ b2;

:: XBOOLE_1:th 97
theorem
for b1, b2, b3 being set
      st b1 \ b2 c= b3 & b2 \ b1 c= b3
   holds b1 \+\ b2 c= b3;

:: XBOOLE_1:th 98
theorem
for b1, b2 being set holds
b1 \/ b2 = b1 \+\ (b2 \ b1);

:: XBOOLE_1:th 99
theorem
for b1, b2, b3 being set holds
(b1 \+\ b2) \ b3 = (b1 \ (b2 \/ b3)) \/ (b2 \ (b1 \/ b3));

:: XBOOLE_1:th 100
theorem
for b1, b2 being set holds
b1 \ b2 = b1 \+\ (b1 /\ b2);

:: XBOOLE_1:th 101
theorem
for b1, b2 being set holds
b1 \+\ b2 = (b1 \/ b2) \ (b1 /\ b2);

:: XBOOLE_1:th 102
theorem
for b1, b2, b3 being set holds
b1 \ (b2 \+\ b3) = (b1 \ (b2 \/ b3)) \/ ((b1 /\ b2) /\ b3);

:: XBOOLE_1:th 103
theorem
for b1, b2 being set holds
b1 /\ b2 misses b1 \+\ b2;

:: XBOOLE_1:th 104
theorem
for b1, b2 being set holds
   (not b1 c< b2 & b1 <> b2 implies b2 c< b1)
iff
   b1,b2 are_c=-comparable;

:: XBOOLE_1:th 105
theorem
for b1, b2 being set
      st b1 c< b2
   holds b2 \ b1 <> {};

:: XBOOLE_1:th 106
theorem
for b1, b2, b3 being set
      st b1 c= b2 \ b3
   holds b1 c= b2 & b1 misses b3;

:: XBOOLE_1:th 107
theorem
for b1, b2, b3 being set holds
   b1 c= b2 \+\ b3
iff
   b1 c= b2 \/ b3 & b1 misses b2 /\ b3;

:: XBOOLE_1:th 108
theorem
for b1, b2, b3 being set
      st b1 c= b2
   holds b1 /\ b3 c= b2;

:: XBOOLE_1:th 109
theorem
for b1, b2, b3 being set
      st b1 c= b2
   holds b1 \ b3 c= b2;

:: XBOOLE_1:th 110
theorem
for b1, b2, b3 being set
      st b1 c= b2 & b3 c= b2
   holds b1 \+\ b3 c= b2;

:: XBOOLE_1:th 111
theorem
for b1, b2, b3 being set holds
(b1 /\ b2) \ (b3 /\ b2) = (b1 \ b3) /\ b2;

:: XBOOLE_1:th 112
theorem
for b1, b2, b3 being set holds
(b1 /\ b2) \+\ (b3 /\ b2) = (b1 \+\ b3) /\ b2;

:: XBOOLE_1:th 113
theorem
for b1, b2, b3, b4 being set holds
((b1 \/ b2) \/ b3) \/ b4 = b1 \/ ((b2 \/ b3) \/ b4);

:: XBOOLE_1:th 114
theorem
for b1, b2, b3, b4 being set
      st b1 misses b4 & b2 misses b4 & b3 misses b4
   holds (b1 \/ b2) \/ b3 misses b4;

:: XBOOLE_1:th 115
theorem
for b1 being set holds
   not b1 c< {};