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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 6672 | . . . . 5 ⊢ (𝐴 = (card‘𝑥) → (card‘𝐴) = (card‘(card‘𝑥))) | |
3 | cardidm 9390 | . . . . 5 ⊢ (card‘(card‘𝑥)) = (card‘𝑥) | |
4 | 2, 3 | syl6eq 2874 | . . . 4 ⊢ (𝐴 = (card‘𝑥) → (card‘𝐴) = (card‘𝑥)) |
5 | 1, 4 | eqtr4d 2861 | . . 3 ⊢ (𝐴 = (card‘𝑥) → 𝐴 = (card‘𝐴)) |
6 | 5 | exlimiv 1931 | . 2 ⊢ (∃𝑥 𝐴 = (card‘𝑥) → 𝐴 = (card‘𝐴)) |
7 | fvex 6685 | . . . 4 ⊢ (card‘𝐴) ∈ V | |
8 | eleq1 2902 | . . . 4 ⊢ (𝐴 = (card‘𝐴) → (𝐴 ∈ V ↔ (card‘𝐴) ∈ V)) | |
9 | 7, 8 | mpbiri 260 | . . 3 ⊢ (𝐴 = (card‘𝐴) → 𝐴 ∈ V) |
10 | fveq2 6672 | . . . . 5 ⊢ (𝑥 = 𝐴 → (card‘𝑥) = (card‘𝐴)) | |
11 | 10 | eqeq2d 2834 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝐴 = (card‘𝑥) ↔ 𝐴 = (card‘𝐴))) |
12 | 11 | spcegv 3599 | . . 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 211 | 1 ⊢ (∃𝑥 𝐴 = (card‘𝑥) ↔ 𝐴 = (card‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1537 ∃wex 1780 ∈ wcel 2114 Vcvv 3496 ‘cfv 6357 cardccrd 9366 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-ord 6196 df-on 6197 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-er 8291 df-en 8512 df-card 9370 |
This theorem is referenced by: iscard4 39907 |
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