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Theorem carden 8141
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 7533). (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 8065 . . . . . 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 7554 . . . . 5  |-  ( A  e.  dom  card  ->  (
card `  A )  ~~  A )
4 ensym 6878 . . . . 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 8065 . . . . . . 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 7554 . . . . . 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 4017 . . . 4  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  ( card `  A )  =  (
card `  B )
)  ->  ( card `  A )  ~~  B
)
12 entr 6881 . . . 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 7568 . 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 3997   dom cdm 4661   ` cfv 4673    ~~ cen 6828   cardccrd 7536
This theorem is referenced by:  cardeq0  8142  ficard  8155  carinttar2  25270
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 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-ac2 8057
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 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-se 4325  df-we 4326  df-ord 4367  df-on 4368  df-suc 4370  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-isom 4690  df-iota 6225  df-riota 6272  df-recs 6356  df-er 6628  df-en 6832  df-card 7540  df-ac 7711
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