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Mirrors > Home > MPE Home > Th. List > oncard | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
oncard | ⊢ (∃𝑥 𝐴 = (card‘𝑥) ↔ 𝐴 = (card‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . . 4 ⊢ (𝐴 = (card‘𝑥) → 𝐴 = (card‘𝑥)) | |
2 | fveq2 6897 | . . . . 5 ⊢ (𝐴 = (card‘𝑥) → (card‘𝐴) = (card‘(card‘𝑥))) | |
3 | cardidm 9983 | . . . . 5 ⊢ (card‘(card‘𝑥)) = (card‘𝑥) | |
4 | 2, 3 | eqtrdi 2784 | . . . 4 ⊢ (𝐴 = (card‘𝑥) → (card‘𝐴) = (card‘𝑥)) |
5 | 1, 4 | eqtr4d 2771 | . . 3 ⊢ (𝐴 = (card‘𝑥) → 𝐴 = (card‘𝐴)) |
6 | 5 | exlimiv 1926 | . 2 ⊢ (∃𝑥 𝐴 = (card‘𝑥) → 𝐴 = (card‘𝐴)) |
7 | fvex 6910 | . . . 4 ⊢ (card‘𝐴) ∈ V | |
8 | eleq1 2817 | . . . 4 ⊢ (𝐴 = (card‘𝐴) → (𝐴 ∈ V ↔ (card‘𝐴) ∈ V)) | |
9 | 7, 8 | mpbiri 258 | . . 3 ⊢ (𝐴 = (card‘𝐴) → 𝐴 ∈ V) |
10 | fveq2 6897 | . . . . 5 ⊢ (𝑥 = 𝐴 → (card‘𝑥) = (card‘𝐴)) | |
11 | 10 | eqeq2d 2739 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝐴 = (card‘𝑥) ↔ 𝐴 = (card‘𝐴))) |
12 | 11 | spcegv 3584 | . . 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 208 | 1 ⊢ (∃𝑥 𝐴 = (card‘𝑥) ↔ 𝐴 = (card‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1534 ∃wex 1774 ∈ wcel 2099 Vcvv 3471 ‘cfv 6548 cardccrd 9959 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-ord 6372 df-on 6373 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-er 8725 df-en 8965 df-card 9963 |
This theorem is referenced by: iscard4 42963 |
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