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Theorem carden 8382
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 the least possible rank (see karden 7775). (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 8306 . . . . . 6  |-  ( A  e.  C  ->  A  e.  dom  card )
21ad2antrr 707 . . . . 5  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  ( card `  A )  =  (
card `  B )
)  ->  A  e.  dom  card )
3 cardid2 7796 . . . . 5  |-  ( A  e.  dom  card  ->  (
card `  A )  ~~  A )
4 ensym 7115 . . . . 5  |-  ( (
card `  A )  ~~  A  ->  A  ~~  ( card `  A )
)
52, 3, 43syl 19 . . . 4  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  ( card `  A )  =  (
card `  B )
)  ->  A  ~~  ( card `  A )
)
6 simpr 448 . . . . 5  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  ( card `  A )  =  (
card `  B )
)  ->  ( card `  A )  =  (
card `  B )
)
7 numth3 8306 . . . . . . 7  |-  ( B  e.  D  ->  B  e.  dom  card )
87ad2antlr 708 . . . . . 6  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  ( card `  A )  =  (
card `  B )
)  ->  B  e.  dom  card )
98cardidd 8380 . . . . 5  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  ( card `  A )  =  (
card `  B )
)  ->  ( card `  B )  ~~  B
)
106, 9eqbrtrd 4192 . . . 4  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  ( card `  A )  =  (
card `  B )
)  ->  ( card `  A )  ~~  B
)
11 entr 7118 . . . 4  |-  ( ( A  ~~  ( card `  A )  /\  ( card `  A )  ~~  B )  ->  A  ~~  B )
125, 10, 11syl2anc 643 . . 3  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  ( card `  A )  =  (
card `  B )
)  ->  A  ~~  B )
1312ex 424 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( ( card `  A
)  =  ( card `  B )  ->  A  ~~  B ) )
14 carden2b 7810 . 2  |-  ( A 
~~  B  ->  ( card `  A )  =  ( card `  B
) )
1513, 14impbid1 195 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( ( card `  A
)  =  ( card `  B )  <->  A  ~~  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   class class class wbr 4172   dom cdm 4837   ` cfv 5413    ~~ cen 7065   cardccrd 7778
This theorem is referenced by:  cardeq0  8383  ficard  8396
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-ac2 8299
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-suc 4547  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6508  df-recs 6592  df-er 6864  df-en 7069  df-card 7782  df-ac 7953
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