Theorem oncard 9889

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

Step	Hyp	Ref	Expression
1		id 22	. . . 4 ⊢ (𝐴 = (card‘𝑥) → 𝐴 = (card‘𝑥))
2		fveq2 6840	. . . . 5 ⊢ (𝐴 = (card‘𝑥) → (card‘𝐴) = (card‘(card‘𝑥)))
3		cardidm 9888	. . . . 5 ⊢ (card‘(card‘𝑥)) = (card‘𝑥)
4	2, 3	eqtrdi 2780	. . . 4 ⊢ (𝐴 = (card‘𝑥) → (card‘𝐴) = (card‘𝑥))
5	1, 4	eqtr4d 2767	. . 3 ⊢ (𝐴 = (card‘𝑥) → 𝐴 = (card‘𝐴))
6	5	exlimiv 1930	. 2 ⊢ (∃𝑥 𝐴 = (card‘𝑥) → 𝐴 = (card‘𝐴))
7		fvex 6853	. . . 4 ⊢ (card‘𝐴) ∈ V
8		eleq1 2816	. . . 4 ⊢ (𝐴 = (card‘𝐴) → (𝐴 ∈ V ↔ (card‘𝐴) ∈ V))
9	7, 8	mpbiri 258	. . 3 ⊢ (𝐴 = (card‘𝐴) → 𝐴 ∈ V)
10		fveq2 6840	. . . . 5 ⊢ (𝑥 = 𝐴 → (card‘𝑥) = (card‘𝐴))
11	10	eqeq2d 2740	. . . 4 ⊢ (𝑥 = 𝐴 → (𝐴 = (card‘𝑥) ↔ 𝐴 = (card‘𝐴)))
12	11	spcegv 3560	. . 3 ⊢ (𝐴 ∈ V → (𝐴 = (card‘𝐴) → ∃𝑥 𝐴 = (card‘𝑥)))
13	9, 12	mpcom 38	. 2 ⊢ (𝐴 = (card‘𝐴) → ∃𝑥 𝐴 = (card‘𝑥))
14	6, 13	impbii 209	1 ⊢ (∃𝑥 𝐴 = (card‘𝑥) ↔ 𝐴 = (card‘𝐴))

Syntax hints:   ↔ wb 206   = wceq 1540  ∃wex 1779   ∈ wcel 2109  Vcvv 3444  ‘cfv 6499  cardccrd 9864

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6323  df-on 6324  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-er 8648  df-en 8896  df-card 9868

This theorem is referenced by:  iscard4  43495