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Theorem oncard 9998
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 6907 . . . . 5 (𝐴 = (card‘𝑥) → (card‘𝐴) = (card‘(card‘𝑥)))
3 cardidm 9997 . . . . 5 (card‘(card‘𝑥)) = (card‘𝑥)
42, 3eqtrdi 2791 . . . 4 (𝐴 = (card‘𝑥) → (card‘𝐴) = (card‘𝑥))
51, 4eqtr4d 2778 . . 3 (𝐴 = (card‘𝑥) → 𝐴 = (card‘𝐴))
65exlimiv 1928 . 2 (∃𝑥 𝐴 = (card‘𝑥) → 𝐴 = (card‘𝐴))
7 fvex 6920 . . . 4 (card‘𝐴) ∈ V
8 eleq1 2827 . . . 4 (𝐴 = (card‘𝐴) → (𝐴 ∈ V ↔ (card‘𝐴) ∈ V))
97, 8mpbiri 258 . . 3 (𝐴 = (card‘𝐴) → 𝐴 ∈ V)
10 fveq2 6907 . . . . 5 (𝑥 = 𝐴 → (card‘𝑥) = (card‘𝐴))
1110eqeq2d 2746 . . . 4 (𝑥 = 𝐴 → (𝐴 = (card‘𝑥) ↔ 𝐴 = (card‘𝐴)))
1211spcegv 3597 . . 3 (𝐴 ∈ V → (𝐴 = (card‘𝐴) → ∃𝑥 𝐴 = (card‘𝑥)))
139, 12mpcom 38 . 2 (𝐴 = (card‘𝐴) → ∃𝑥 𝐴 = (card‘𝑥))
146, 13impbii 209 1 (∃𝑥 𝐴 = (card‘𝑥) ↔ 𝐴 = (card‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1537  wex 1776  wcel 2106  Vcvv 3478  cfv 6563  cardccrd 9973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-er 8744  df-en 8985  df-card 9977
This theorem is referenced by:  iscard4  43523
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