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Theorem oncard 7477
Description: A set is a cardinal number iff it equals its own cardinal number. Proposition 10.9 of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 7-Jan-2013.)
Assertion
Ref Expression
oncard  |-  ( E. x  A  =  (
card `  x )  <->  A  =  ( card `  A
) )
Distinct variable group:    x, A

Proof of Theorem oncard
StepHypRef Expression
1 id 21 . . . 4  |-  ( A  =  ( card `  x
)  ->  A  =  ( card `  x )
)
2 fveq2 5377 . . . . 5  |-  ( A  =  ( card `  x
)  ->  ( card `  A )  =  (
card `  ( card `  x ) ) )
3 cardidm 7476 . . . . 5  |-  ( card `  ( card `  x
) )  =  (
card `  x )
42, 3syl6eq 2301 . . . 4  |-  ( A  =  ( card `  x
)  ->  ( card `  A )  =  (
card `  x )
)
51, 4eqtr4d 2288 . . 3  |-  ( A  =  ( card `  x
)  ->  A  =  ( card `  A )
)
65exlimiv 2023 . 2  |-  ( E. x  A  =  (
card `  x )  ->  A  =  ( card `  A ) )
7 fvex 5391 . . . 4  |-  ( card `  A )  e.  _V
8 eleq1 2313 . . . 4  |-  ( A  =  ( card `  A
)  ->  ( A  e.  _V  <->  ( card `  A
)  e.  _V )
)
97, 8mpbiri 226 . . 3  |-  ( A  =  ( card `  A
)  ->  A  e.  _V )
10 fveq2 5377 . . . . 5  |-  ( x  =  A  ->  ( card `  x )  =  ( card `  A
) )
1110eqeq2d 2264 . . . 4  |-  ( x  =  A  ->  ( A  =  ( card `  x )  <->  A  =  ( card `  A )
) )
1211cla4egv 2806 . . 3  |-  ( A  e.  _V  ->  ( A  =  ( card `  A )  ->  E. x  A  =  ( card `  x ) ) )
139, 12mpcom 34 . 2  |-  ( A  =  ( card `  A
)  ->  E. x  A  =  ( card `  x ) )
146, 13impbii 182 1  |-  ( E. x  A  =  (
card `  x )  <->  A  =  ( card `  A
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178   E.wex 1537    = wceq 1619    e. wcel 1621   _Vcvv 2727   ` cfv 4592   cardccrd 7452
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-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
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-rab 2516  df-v 2729  df-sbc 2922  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-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-we 4247  df-ord 4288  df-on 4289  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-er 6546  df-en 6750  df-card 7456
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