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Theorem oncard 7683
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 19 . . . 4  |-  ( A  =  ( card `  x
)  ->  A  =  ( card `  x )
)
2 fveq2 5608 . . . . 5  |-  ( A  =  ( card `  x
)  ->  ( card `  A )  =  (
card `  ( card `  x ) ) )
3 cardidm 7682 . . . . 5  |-  ( card `  ( card `  x
) )  =  (
card `  x )
42, 3syl6eq 2406 . . . 4  |-  ( A  =  ( card `  x
)  ->  ( card `  A )  =  (
card `  x )
)
51, 4eqtr4d 2393 . . 3  |-  ( A  =  ( card `  x
)  ->  A  =  ( card `  A )
)
65exlimiv 1634 . 2  |-  ( E. x  A  =  (
card `  x )  ->  A  =  ( card `  A ) )
7 fvex 5622 . . . 4  |-  ( card `  A )  e.  _V
8 eleq1 2418 . . . 4  |-  ( A  =  ( card `  A
)  ->  ( A  e.  _V  <->  ( card `  A
)  e.  _V )
)
97, 8mpbiri 224 . . 3  |-  ( A  =  ( card `  A
)  ->  A  e.  _V )
10 fveq2 5608 . . . . 5  |-  ( x  =  A  ->  ( card `  x )  =  ( card `  A
) )
1110eqeq2d 2369 . . . 4  |-  ( x  =  A  ->  ( A  =  ( card `  x )  <->  A  =  ( card `  A )
) )
1211spcegv 2945 . . 3  |-  ( A  e.  _V  ->  ( A  =  ( card `  A )  ->  E. x  A  =  ( card `  x ) ) )
139, 12mpcom 32 . 2  |-  ( A  =  ( card `  A
)  ->  E. x  A  =  ( card `  x ) )
146, 13impbii 180 1  |-  ( E. x  A  =  (
card `  x )  <->  A  =  ( card `  A
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   E.wex 1541    = wceq 1642    e. wcel 1710   _Vcvv 2864   ` cfv 5337   cardccrd 7658
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-er 6747  df-en 6952  df-card 7662
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