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Theorem carden 8415
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 7808). (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 8339 . . . . . 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 7829 . . . . 5  |-  ( A  e.  dom  card  ->  (
card `  A )  ~~  A )
4 ensym 7147 . . . . 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 8339 . . . . . . 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 8413 . . . . 5  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  ( card `  A )  =  (
card `  B )
)  ->  ( card `  B )  ~~  B
)
106, 9eqbrtrd 4224 . . . 4  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  ( card `  A )  =  (
card `  B )
)  ->  ( card `  A )  ~~  B
)
11 entr 7150 . . . 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 7843 . 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 1652    e. wcel 1725   class class class wbr 4204   dom cdm 4869   ` cfv 5445    ~~ cen 7097   cardccrd 7811
This theorem is referenced by:  cardeq0  8416  ficard  8429
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-ac2 8332
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-suc 4579  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-isom 5454  df-riota 6540  df-recs 6624  df-er 6896  df-en 7101  df-card 7815  df-ac 7986
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