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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 6884 | . . . . 5 ⊢ (𝐴 = (card‘𝑥) → (card‘𝐴) = (card‘(card‘𝑥))) | |
3 | cardidm 9953 | . . . . 5 ⊢ (card‘(card‘𝑥)) = (card‘𝑥) | |
4 | 2, 3 | eqtrdi 2782 | . . . 4 ⊢ (𝐴 = (card‘𝑥) → (card‘𝐴) = (card‘𝑥)) |
5 | 1, 4 | eqtr4d 2769 | . . 3 ⊢ (𝐴 = (card‘𝑥) → 𝐴 = (card‘𝐴)) |
6 | 5 | exlimiv 1925 | . 2 ⊢ (∃𝑥 𝐴 = (card‘𝑥) → 𝐴 = (card‘𝐴)) |
7 | fvex 6897 | . . . 4 ⊢ (card‘𝐴) ∈ V | |
8 | eleq1 2815 | . . . 4 ⊢ (𝐴 = (card‘𝐴) → (𝐴 ∈ V ↔ (card‘𝐴) ∈ V)) | |
9 | 7, 8 | mpbiri 258 | . . 3 ⊢ (𝐴 = (card‘𝐴) → 𝐴 ∈ V) |
10 | fveq2 6884 | . . . . 5 ⊢ (𝑥 = 𝐴 → (card‘𝑥) = (card‘𝐴)) | |
11 | 10 | eqeq2d 2737 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝐴 = (card‘𝑥) ↔ 𝐴 = (card‘𝐴))) |
12 | 11 | spcegv 3581 | . . 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 1533 ∃wex 1773 ∈ wcel 2098 Vcvv 3468 ‘cfv 6536 cardccrd 9929 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-ord 6360 df-on 6361 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-er 8702 df-en 8939 df-card 9933 |
This theorem is referenced by: iscard4 42841 |
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