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Theorem oncard 8142
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 22 . . . 4  |-  ( A  =  ( card `  x
)  ->  A  =  ( card `  x )
)
2 fveq2 5703 . . . . 5  |-  ( A  =  ( card `  x
)  ->  ( card `  A )  =  (
card `  ( card `  x ) ) )
3 cardidm 8141 . . . . 5  |-  ( card `  ( card `  x
) )  =  (
card `  x )
42, 3syl6eq 2491 . . . 4  |-  ( A  =  ( card `  x
)  ->  ( card `  A )  =  (
card `  x )
)
51, 4eqtr4d 2478 . . 3  |-  ( A  =  ( card `  x
)  ->  A  =  ( card `  A )
)
65exlimiv 1688 . 2  |-  ( E. x  A  =  (
card `  x )  ->  A  =  ( card `  A ) )
7 fvex 5713 . . . 4  |-  ( card `  A )  e.  _V
8 eleq1 2503 . . . 4  |-  ( A  =  ( card `  A
)  ->  ( A  e.  _V  <->  ( card `  A
)  e.  _V )
)
97, 8mpbiri 233 . . 3  |-  ( A  =  ( card `  A
)  ->  A  e.  _V )
10 fveq2 5703 . . . . 5  |-  ( x  =  A  ->  ( card `  x )  =  ( card `  A
) )
1110eqeq2d 2454 . . . 4  |-  ( x  =  A  ->  ( A  =  ( card `  x )  <->  A  =  ( card `  A )
) )
1211spcegv 3070 . . 3  |-  ( A  e.  _V  ->  ( A  =  ( card `  A )  ->  E. x  A  =  ( card `  x ) ) )
139, 12mpcom 36 . 2  |-  ( A  =  ( card `  A
)  ->  E. x  A  =  ( card `  x ) )
146, 13impbii 188 1  |-  ( E. x  A  =  (
card `  x )  <->  A  =  ( card `  A
) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1369   E.wex 1586    e. wcel 1756   _Vcvv 2984   ` cfv 5430   cardccrd 8117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-sbc 3199  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-er 7113  df-en 7323  df-card 8121
This theorem is referenced by: (None)
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