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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 6843 | . . . . 5 ⊢ (𝐴 = (card‘𝑥) → (card‘𝐴) = (card‘(card‘𝑥))) | |
3 | cardidm 9900 | . . . . 5 ⊢ (card‘(card‘𝑥)) = (card‘𝑥) | |
4 | 2, 3 | eqtrdi 2789 | . . . 4 ⊢ (𝐴 = (card‘𝑥) → (card‘𝐴) = (card‘𝑥)) |
5 | 1, 4 | eqtr4d 2776 | . . 3 ⊢ (𝐴 = (card‘𝑥) → 𝐴 = (card‘𝐴)) |
6 | 5 | exlimiv 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)) | |
9 | 7, 8 | mpbiri 258 | . . 3 ⊢ (𝐴 = (card‘𝐴) → 𝐴 ∈ V) |
10 | fveq2 6843 | . . . . 5 ⊢ (𝑥 = 𝐴 → (card‘𝑥) = (card‘𝐴)) | |
11 | 10 | eqeq2d 2744 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝐴 = (card‘𝑥) ↔ 𝐴 = (card‘𝐴))) |
12 | 11 | spcegv 3555 | . . 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 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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