Article CARD_4, MML version 4.99.1005

:: CARD_4:th 1
theorem
for b1 being set holds
      b1 is finite
   iff
      Card b1 is finite;

:: CARD_4:th 2
theorem
for b1 being set holds
      b1 is finite
   iff
      Card b1 in alef 0;

:: CARD_4:th 3
theorem
for b1 being set
      st b1 is finite
   holds Card b1 in alef 0 & Card b1 in omega;

:: CARD_4:th 4
theorem
for b1 being set holds
      b1 is finite
   iff
      ex b2 being Element of NAT st
         Card b1 = Card b2;

:: CARD_4:th 8
theorem
for b1 being ordinal set holds
      b1 is infinite
   iff
      omega c= b1;

:: CARD_4:th 9
theorem
for b1 being cardinal set holds
      b1 is finite
   iff
      b1 in alef 0;

:: CARD_4:th 11
theorem
for b1 being cardinal set holds
      b1 is infinite
   iff
      alef 0 c= b1;

:: CARD_4:th 13
theorem
for b1, b2 being cardinal set
      st b1 is finite & b2 is infinite
   holds b1 in b2 & b1 c= b2;

:: CARD_4:th 14
theorem
for b1 being set holds
      b1 is infinite
   iff
      ex b2 being set st
         b2 c= b1 & Card b2 = alef 0;

:: CARD_4:th 15
theorem
NAT is infinite;

:: CARD_4:th 17
theorem
for b1 being set holds
      b1 = {}
   iff
      Card b1 = 0;

:: CARD_4:th 19
theorem
for b1 being cardinal set holds
   0 c= b1;

:: CARD_4:th 20
theorem
for b1, b2 being set holds
   Card b1 = Card b2
iff
   nextcard b1 = nextcard b2;

:: CARD_4:th 21
theorem
for b1, b2 being cardinal set holds
   b1 = b2
iff
   nextcard b2 = nextcard b1;

:: CARD_4:th 22
theorem
for b1, b2 being cardinal set holds
   b1 in b2
iff
   nextcard b1 c= b2;

:: CARD_4:th 23
theorem
for b1, b2 being cardinal set holds
   b1 in nextcard b2
iff
   b1 c= b2;

:: CARD_4:th 24
theorem
for b1 being cardinal set holds
      0 in b1
   iff
      1 c= b1;

:: CARD_4:th 25
theorem
for b1 being cardinal set holds
      1 in b1
   iff
      2 c= b1;

:: CARD_4:th 26
theorem
for b1, b2 being cardinal set
      st b1 is finite & (b2 c= b1 or b2 in b1)
   holds b2 is finite;

:: CARD_4:th 27
theorem
for b1 being ordinal set holds
      b1 is being_limit_ordinal
   iff
      for b2 being ordinal set
      for b3 being Element of NAT
            st b2 in b1
         holds b2 +^ b3 in b1;

:: CARD_4:th 28
theorem
for b1 being Element of NAT
for b2 being ordinal set holds
   b2 +^ succ b1 = (succ b2) +^ b1 &
    b2 +^ (b1 + 1) = (succ b2) +^ b1;

:: CARD_4:th 29
theorem
for b1 being ordinal set holds
   ex b2 being Element of NAT st
      b1 *^ succ 1 = b1 +^ b2;

:: CARD_4:th 30
theorem
for b1 being ordinal set
      st b1 is being_limit_ordinal
   holds b1 *^ succ 1 = b1;

:: CARD_4:th 31
theorem
for b1 being ordinal set
      st omega c= b1
   holds 1 +^ b1 = b1;

:: CARD_4:th 32
theorem
for b1 being cardinal set
      st b1 is infinite
   holds b1 is being_limit_ordinal;

:: CARD_4:th 33
theorem
for b1 being cardinal set
      st b1 is infinite
   holds b1 +` b1 = b1;

:: CARD_4:th 34
theorem
for b1, b2 being cardinal set
      st b1 is infinite & (b2 c= b1 or b2 in b1)
   holds b1 +` b2 = b1 & b2 +` b1 = b1;

:: CARD_4:th 35
theorem
for b1, b2 being set
      st b1 is infinite & (b1,b2 are_equipotent or b2,b1 are_equipotent)
   holds b1 \/ b2,b1 are_equipotent & Card (b1 \/ b2) = Card b1;

:: CARD_4:th 36
theorem
for b1, b2 being set
      st b1 is infinite & b2 is finite
   holds b1 \/ b2,b1 are_equipotent & Card (b1 \/ b2) = Card b1;

:: CARD_4:th 37
theorem
for b1, b2 being set
      st b1 is infinite &
         (Card b2 in Card b1 or Card b2 c= Card b1)
   holds b1 \/ b2,b1 are_equipotent & Card (b1 \/ b2) = Card b1;

:: CARD_4:th 38
theorem
for b1, b2 being finite cardinal set holds
b1 +` b2 is finite;

:: CARD_4:th 39
theorem
for b1, b2 being cardinal set
      st b1 is infinite
   holds b1 +` b2 is infinite & b2 +` b1 is infinite;

:: CARD_4:th 40
theorem
for b1, b2 being finite cardinal set holds
b1 *` b2 is finite;

:: CARD_4:th 41
theorem
for b1, b2, b3, b4 being cardinal set
      st ((b1 in b2 implies not b3 in b4) & (b1 c= b2 implies not b3 in b4) & (b1 in b2 implies not b3 c= b4) implies b1 c= b2 & b3 c= b4)
   holds b1 +` b3 c= b2 +` b4 & b3 +` b1 c= b2 +` b4;

:: CARD_4:th 42
theorem
for b1, b2, b3 being cardinal set
      st (b1 in b2 or b1 c= b2)
   holds b3 +` b1 c= b3 +` b2 & b3 +` b1 c= b2 +` b3 & b1 +` b3 c= b3 +` b2 & b1 +` b3 c= b2 +` b3;

:: CARD_4:attrnot 1 => CARD_4:attr 1
definition
  let a1 be set;
  attr a1 is countable means
    Card a1 c= alef 0;
end;

:: CARD_4:dfs 1
definiens
  let a1 be set;
To prove
     a1 is countable
it is sufficient to prove
  thus Card a1 c= alef 0;

:: CARD_4:def 1
theorem
for b1 being set holds
      b1 is countable
   iff
      Card b1 c= alef 0;

:: CARD_4:th 43
theorem
for b1 being set
      st b1 is finite
   holds b1 is countable;

:: CARD_4:th 44
theorem
omega is countable & NAT is countable;

:: CARD_4:th 45
theorem
for b1 being set holds
      b1 is countable
   iff
      ex b2 being Relation-like Function-like set st
         proj1 b2 = NAT & b1 c= proj2 b2;

:: CARD_4:th 46
theorem
for b1, b2 being set
      st b1 c= b2 & b2 is countable
   holds b1 is countable;

:: CARD_4:th 47
theorem
for b1, b2 being set
      st b1 is countable & b2 is countable
   holds b1 \/ b2 is countable;

:: CARD_4:th 48
theorem
for b1, b2 being set
      st b1 is countable
   holds b1 /\ b2 is countable & b2 /\ b1 is countable;

:: CARD_4:th 49
theorem
for b1, b2 being set
      st b1 is countable
   holds b1 \ b2 is countable;

:: CARD_4:th 50
theorem
for b1, b2 being set
      st b1 is countable & b2 is countable
   holds b1 \+\ b2 is countable;

:: CARD_4:th 51
theorem
for b1 being Element of NAT
for b2 being Element of REAL holds
      (b2 = 0 implies b1 = 0)
   iff
      b2 |^ b1 <> 0;

:: CARD_4:th 52
theorem
for b1, b2, b3, b4 being Element of NAT
      st (2 |^ b1) * ((2 * b2) + 1) = (2 |^ b3) * ((2 * b4) + 1)
   holds b1 = b3 & b2 = b4;

:: CARD_4:th 53
theorem
[:NAT,NAT:],NAT are_equipotent &
 Card NAT = Card [:NAT,NAT:];

:: CARD_4:th 54
theorem
(alef 0) *` alef 0 = alef 0;

:: CARD_4:th 55
theorem
for b1, b2 being set
      st b1 is countable & b2 is countable
   holds [:b1,b2:] is countable;

:: CARD_4:th 56
theorem
for b1 being non empty set holds
   1 -tuples_on b1,b1 are_equipotent & Card (1 -tuples_on b1) = Card b1;

:: CARD_4:th 57
theorem
for b1 being non empty set
for b2, b3 being Element of NAT holds
[:b2 -tuples_on b1,b3 -tuples_on b1:],(b2 + b3) -tuples_on b1 are_equipotent &
 Card [:b2 -tuples_on b1,b3 -tuples_on b1:] = Card ((b2 + b3) -tuples_on b1);

:: CARD_4:th 58
theorem
for b1 being non empty set
for b2 being Element of NAT
      st b1 is countable
   holds b2 -tuples_on b1 is countable;

:: CARD_4:th 59
theorem
for b1, b2 being cardinal set
for b3 being Relation-like Function-like set
      st Card proj1 b3 c= b1 &
         (for b4 being set
               st b4 in proj1 b3
            holds Card (b3 . b4) c= b2)
   holds Card Union b3 c= b1 *` b2;

:: CARD_4:th 60
theorem
for b1 being set
for b2, b3 being cardinal set
      st Card b1 c= b2 &
         (for b4 being set
               st b4 in b1
            holds Card b4 c= b3)
   holds Card union b1 c= b2 *` b3;

:: CARD_4:th 61
theorem
for b1 being Relation-like Function-like set
      st proj1 b1 is countable &
         (for b2 being set
               st b2 in proj1 b1
            holds b1 . b2 is countable)
   holds Union b1 is countable;

:: CARD_4:th 62
theorem
for b1 being set
      st b1 is countable &
         (for b2 being set
               st b2 in b1
            holds b2 is countable)
   holds union b1 is countable;

:: CARD_4:th 63
theorem
for b1 being Relation-like Function-like set
      st proj1 b1 is finite &
         (for b2 being set
               st b2 in proj1 b1
            holds b1 . b2 is finite)
   holds Union b1 is finite;

:: CARD_4:th 65
theorem
for b1 being non empty set
      st b1 is countable
   holds b1 * is countable;

:: CARD_4:th 66
theorem
for b1 being non empty set holds
   alef 0 c= Card (b1 *);

:: CARD_4:sch 1
scheme CARD_4:sch 1
{F1 -> set}:
{F1(b1) where b1 is Element of NAT: P1[b1]} is countable


:: CARD_4:sch 2
scheme CARD_4:sch 2
{F1 -> set}:
{F1(b1, b2) where b1 is Element of NAT, b2 is Element of NAT: P1[b1, b2]} is countable


:: CARD_4:sch 3
scheme CARD_4:sch 3
{F1 -> set}:
{F1(b1, b2, b3) where b1 is Element of NAT, b2 is Element of NAT, b3 is Element of NAT: P1[b1, b2, b3]} is countable


:: CARD_4:th 67
theorem
for b1 being Element of NAT holds
   (alef 0) *` Card b1 c= alef 0 &
    (Card b1) *` alef 0 c= alef 0;

:: CARD_4:th 68
theorem
for b1, b2, b3, b4 being cardinal set
      st ((b1 in b2 implies not b3 in b4) & (b1 c= b2 implies not b3 in b4) & (b1 in b2 implies not b3 c= b4) implies b1 c= b2 & b3 c= b4)
   holds b1 *` b3 c= b2 *` b4 & b3 *` b1 c= b2 *` b4;

:: CARD_4:th 69
theorem
for b1, b2, b3 being cardinal set
      st (b1 in b2 or b1 c= b2)
   holds b3 *` b1 c= b3 *` b2 & b3 *` b1 c= b2 *` b3 & b1 *` b3 c= b3 *` b2 & b1 *` b3 c= b2 *` b3;

:: CARD_4:th 70
theorem
for b1, b2, b3, b4 being cardinal set
      st ((b1 in b2 implies not b3 in b4) & (b1 c= b2 implies not b3 in b4) & (b1 in b2 implies not b3 c= b4) implies b1 c= b2 & b3 c= b4) &
         b1 <> 0
   holds exp(b1,b3) c= exp(b2,b4);

:: CARD_4:th 71
theorem
for b1, b2, b3 being cardinal set
      st (b1 in b2 or b1 c= b2) & b3 <> 0
   holds exp(b3,b1) c= exp(b3,b2) & exp(b1,b3) c= exp(b2,b3);

:: CARD_4:th 72
theorem
for b1, b2 being cardinal set holds
b1 c= b1 +` b2 & b2 c= b1 +` b2;

:: CARD_4:th 73
theorem
for b1, b2 being cardinal set
      st b1 <> 0
   holds b2 c= b2 *` b1 & b2 c= b1 *` b2;

:: CARD_4:th 74
theorem
for b1, b2, b3, b4 being cardinal set
      st b1 in b2 & b3 in b4
   holds b1 +` b3 in b2 +` b4 & b3 +` b1 in b2 +` b4;

:: CARD_4:th 75
theorem
for b1, b2, b3 being cardinal set
      st b1 +` b2 in b1 +` b3
   holds b2 in b3;

:: CARD_4:th 76
theorem
for b1, b2 being set
      st (Card b1) +` Card b2 = Card b1 & Card b2 in Card b1
   holds Card (b1 \ b2) = Card b1;

:: CARD_4:th 77
theorem
for b1 being cardinal set
      st b1 is infinite
   holds b1 *` b1 = b1;

:: CARD_4:th 78
theorem
for b1, b2 being cardinal set
      st b1 is infinite & 0 in b2 & (b2 c= b1 or b2 in b1)
   holds b1 *` b2 = b1 & b2 *` b1 = b1;

:: CARD_4:th 79
theorem
for b1, b2 being cardinal set
      st b1 is infinite & (b2 c= b1 or b2 in b1)
   holds b1 *` b2 c= b1 & b2 *` b1 c= b1;

:: CARD_4:th 80
theorem
for b1 being set
      st b1 is infinite
   holds [:b1,b1:],b1 are_equipotent & Card [:b1,b1:] = Card b1;

:: CARD_4:th 81
theorem
for b1, b2 being set
      st b1 is infinite & b2 is finite & b2 <> {}
   holds [:b1,b2:],b1 are_equipotent & Card [:b1,b2:] = Card b1;

:: CARD_4:th 82
theorem
for b1, b2, b3, b4 being cardinal set
      st b1 in b2 & b3 in b4
   holds b1 *` b3 in b2 *` b4 & b3 *` b1 in b2 *` b4;

:: CARD_4:th 83
theorem
for b1, b2, b3 being cardinal set
      st b1 *` b2 in b1 *` b3
   holds b2 in b3;

:: CARD_4:th 84
theorem
for b1 being set
      st b1 is infinite
   holds Card b1 = (alef 0) *` Card b1;

:: CARD_4:th 85
theorem
for b1, b2 being set
      st b1 <> {} & b1 is finite & b2 is infinite
   holds (Card b2) *` Card b1 = Card b2;

:: CARD_4:th 86
theorem
for b1 being non empty set
for b2 being Element of NAT
      st b1 is infinite & b2 <> 0
   holds b2 -tuples_on b1,b1 are_equipotent & Card (b2 -tuples_on b1) = Card b1;

:: CARD_4:th 87
theorem
for b1 being non empty set
      st b1 is infinite
   holds Card b1 = Card (b1 *);

:: CARD_4:funcreg 1
registration
  let a1 be finite set;
  let a2 be set;
  let a3 be Function-like quasi_total Relation of a1,Fin a2;
  cluster Union a3 -> finite;
end;