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Theorem oncard 9901
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 (∃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 6843 . . . . 5 (𝐴 = (card‘𝑥) → (card‘𝐴) = (card‘(card‘𝑥)))
3 cardidm 9900 . . . . 5 (card‘(card‘𝑥)) = (card‘𝑥)
42, 3eqtrdi 2789 . . . 4 (𝐴 = (card‘𝑥) → (card‘𝐴) = (card‘𝑥))
51, 4eqtr4d 2776 . . 3 (𝐴 = (card‘𝑥) → 𝐴 = (card‘𝐴))
65exlimiv 1934 . 2 (∃𝑥 𝐴 = (card‘𝑥) → 𝐴 = (card‘𝐴))
7 fvex 6856 . . . 4 (card‘𝐴) ∈ V
8 eleq1 2822 . . . 4 (𝐴 = (card‘𝐴) → (𝐴 ∈ V ↔ (card‘𝐴) ∈ V))
97, 8mpbiri 258 . . 3 (𝐴 = (card‘𝐴) → 𝐴 ∈ V)
10 fveq2 6843 . . . . 5 (𝑥 = 𝐴 → (card‘𝑥) = (card‘𝐴))
1110eqeq2d 2744 . . . 4 (𝑥 = 𝐴 → (𝐴 = (card‘𝑥) ↔ 𝐴 = (card‘𝐴)))
1211spcegv 3555 . . 3 (𝐴 ∈ V → (𝐴 = (card‘𝐴) → ∃𝑥 𝐴 = (card‘𝑥)))
139, 12mpcom 38 . 2 (𝐴 = (card‘𝐴) → ∃𝑥 𝐴 = (card‘𝑥))
146, 13impbii 208 1 (∃𝑥 𝐴 = (card‘𝑥) ↔ 𝐴 = (card‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1542  wex 1782  wcel 2107  Vcvv 3444  cfv 6497  cardccrd 9876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ord 6321  df-on 6322  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-er 8651  df-en 8887  df-card 9880
This theorem is referenced by:  iscard4  41893
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