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Theorem oncard 7589
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 5486 . . . . 5  |-  ( A  =  ( card `  x
)  ->  ( card `  A )  =  (
card `  ( card `  x ) ) )
3 cardidm 7588 . . . . 5  |-  ( card `  ( card `  x
) )  =  (
card `  x )
42, 3syl6eq 2333 . . . 4  |-  ( A  =  ( card `  x
)  ->  ( card `  A )  =  (
card `  x )
)
51, 4eqtr4d 2320 . . 3  |-  ( A  =  ( card `  x
)  ->  A  =  ( card `  A )
)
65exlimiv 1667 . 2  |-  ( E. x  A  =  (
card `  x )  ->  A  =  ( card `  A ) )
7 fvex 5500 . . . 4  |-  ( card `  A )  e.  _V
8 eleq1 2345 . . . 4  |-  ( A  =  ( card `  A
)  ->  ( A  e.  _V  <->  ( card `  A
)  e.  _V )
)
97, 8mpbiri 226 . . 3  |-  ( A  =  ( card `  A
)  ->  A  e.  _V )
10 fveq2 5486 . . . . 5  |-  ( x  =  A  ->  ( card `  x )  =  ( card `  A
) )
1110eqeq2d 2296 . . . 4  |-  ( x  =  A  ->  ( A  =  ( card `  x )  <->  A  =  ( card `  A )
) )
1211spcegv 2871 . . 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 1529    = wceq 1624    e. wcel 1685   _Vcvv 2790   ` cfv 5222   cardccrd 7564
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 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512
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 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-er 6656  df-en 6860  df-card 7568
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