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Theorem carden 8168
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 7560). (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 8092 . . . . . 6  |-  ( A  e.  C  ->  A  e.  dom  card )
21ad2antrr 708 . . . . 5  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  ( card `  A )  =  (
card `  B )
)  ->  A  e.  dom  card )
3 cardid2 7581 . . . . 5  |-  ( A  e.  dom  card  ->  (
card `  A )  ~~  A )
4 ensym 6905 . . . . 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 8092 . . . . . . 7  |-  ( B  e.  D  ->  B  e.  dom  card )
87ad2antlr 709 . . . . . 6  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  ( card `  A )  =  (
card `  B )
)  ->  B  e.  dom  card )
9 cardid2 7581 . . . . . 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 4044 . . . 4  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  ( card `  A )  =  (
card `  B )
)  ->  ( card `  A )  ~~  B
)
12 entr 6908 . . . 4  |-  ( ( A  ~~  ( card `  A )  /\  ( card `  A )  ~~  B )  ->  A  ~~  B )
135, 11, 12syl2anc 644 . . 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 7595 . 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 1624    e. wcel 1685   class class class wbr 4024   dom cdm 4688   ` cfv 5221    ~~ cen 6855   cardccrd 7563
This theorem is referenced by:  cardeq0  8169  ficard  8182  carinttar2  25302
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-ac2 8084
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-se 4352  df-we 4353  df-ord 4394  df-on 4395  df-suc 4397  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-isom 5230  df-iota 6252  df-riota 6299  df-recs 6383  df-er 6655  df-en 6859  df-card 7567  df-ac 7738
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