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Theorem carden 8055
Description: Two sets are equinumerous iff their cardinal numbers are equal. This important theorem expresses the essential concept behind "cardinality" or "size." This theorem appears as Proposition 10.10 of [TakeutiZaring] p. 85, Theorem 7P of [Enderton] p. 197, and Theorem 9 of [Suppes] p. 242 (among others). The Axiom of Choice is required for its proof.

The theory of cardinality can also be developed without AC by introducing "card" as a primitive notion and stating this theorem as an axiom, as is done with the axiom for cardinal numbers in [Suppes] p. 111. Finally, if we allow the Axiom of Regularity, we can avoid AC by defining the cardinal number of a set as the set of all sets equinumerous to it and having least possible rank (see karden 7449). (Contributed by NM, 22-Oct-2003.)

Assertion
Ref Expression
carden  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( ( card `  A
)  =  ( card `  B )  <->  A  ~~  B ) )

Proof of Theorem carden
StepHypRef Expression
1 numth3 7981 . . . . . 6  |-  ( A  e.  C  ->  A  e.  dom  card )
21ad2antrr 709 . . . . 5  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  ( card `  A )  =  (
card `  B )
)  ->  A  e.  dom  card )
3 cardid2 7470 . . . . 5  |-  ( A  e.  dom  card  ->  (
card `  A )  ~~  A )
4 ensym 6796 . . . . 5  |-  ( (
card `  A )  ~~  A  ->  A  ~~  ( card `  A )
)
52, 3, 43syl 20 . . . 4  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  ( card `  A )  =  (
card `  B )
)  ->  A  ~~  ( card `  A )
)
6 simpr 449 . . . . 5  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  ( card `  A )  =  (
card `  B )
)  ->  ( card `  A )  =  (
card `  B )
)
7 numth3 7981 . . . . . . 7  |-  ( B  e.  D  ->  B  e.  dom  card )
87ad2antlr 710 . . . . . 6  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  ( card `  A )  =  (
card `  B )
)  ->  B  e.  dom  card )
9 cardid2 7470 . . . . . 6  |-  ( B  e.  dom  card  ->  (
card `  B )  ~~  B )
108, 9syl 17 . . . . 5  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  ( card `  A )  =  (
card `  B )
)  ->  ( card `  B )  ~~  B
)
116, 10eqbrtrd 3940 . . . 4  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  ( card `  A )  =  (
card `  B )
)  ->  ( card `  A )  ~~  B
)
12 entr 6798 . . . 4  |-  ( ( A  ~~  ( card `  A )  /\  ( card `  A )  ~~  B )  ->  A  ~~  B )
135, 11, 12syl2anc 645 . . 3  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  ( card `  A )  =  (
card `  B )
)  ->  A  ~~  B )
1413ex 425 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( ( card `  A
)  =  ( card `  B )  ->  A  ~~  B ) )
15 carden2b 7484 . 2  |-  ( A 
~~  B  ->  ( card `  A )  =  ( card `  B
) )
1614, 15impbid1 196 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( ( card `  A
)  =  ( card `  B )  <->  A  ~~  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621   class class class wbr 3920   dom cdm 4580   ` cfv 4592    ~~ cen 6746   cardccrd 7452
This theorem is referenced by:  cardeq0  8056  ficard  8069  carinttar2  25069
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-ac2 7973
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-se 4246  df-we 4247  df-ord 4288  df-on 4289  df-suc 4291  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-isom 4609  df-iota 6143  df-riota 6190  df-recs 6274  df-er 6546  df-en 6750  df-card 7456  df-ac 7627
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