Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 6103 | . . . . 5 ⊢ (𝐴 = (card‘𝑥) → (card‘𝐴) = (card‘(card‘𝑥))) | |
3 | cardidm 8668 | . . . . 5 ⊢ (card‘(card‘𝑥)) = (card‘𝑥) | |
4 | 2, 3 | syl6eq 2660 | . . . 4 ⊢ (𝐴 = (card‘𝑥) → (card‘𝐴) = (card‘𝑥)) |
5 | 1, 4 | eqtr4d 2647 | . . 3 ⊢ (𝐴 = (card‘𝑥) → 𝐴 = (card‘𝐴)) |
6 | 5 | exlimiv 1845 | . 2 ⊢ (∃𝑥 𝐴 = (card‘𝑥) → 𝐴 = (card‘𝐴)) |
7 | fvex 6113 | . . . 4 ⊢ (card‘𝐴) ∈ V | |
8 | eleq1 2676 | . . . 4 ⊢ (𝐴 = (card‘𝐴) → (𝐴 ∈ V ↔ (card‘𝐴) ∈ V)) | |
9 | 7, 8 | mpbiri 247 | . . 3 ⊢ (𝐴 = (card‘𝐴) → 𝐴 ∈ V) |
10 | fveq2 6103 | . . . . 5 ⊢ (𝑥 = 𝐴 → (card‘𝑥) = (card‘𝐴)) | |
11 | 10 | eqeq2d 2620 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝐴 = (card‘𝑥) ↔ 𝐴 = (card‘𝐴))) |
12 | 11 | spcegv 3267 | . . 3 ⊢ (𝐴 ∈ V → (𝐴 = (card‘𝐴) → ∃𝑥 𝐴 = (card‘𝑥))) |
13 | 9, 12 | mpcom 37 | . 2 ⊢ (𝐴 = (card‘𝐴) → ∃𝑥 𝐴 = (card‘𝑥)) |
14 | 6, 13 | impbii 198 | 1 ⊢ (∃𝑥 𝐴 = (card‘𝑥) ↔ 𝐴 = (card‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 ∃wex 1695 ∈ wcel 1977 Vcvv 3173 ‘cfv 5804 cardccrd 8644 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-ord 5643 df-on 5644 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-er 7629 df-en 7842 df-card 8648 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |