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| Mirrors > Home > MPE Home > Th. List > fineqv | Structured version Visualization version GIF version | ||
| Description: If the Axiom of Infinity is denied, then all sets are finite (which implies the Axiom of Choice). (Contributed by Mario Carneiro, 20-Jan-2013.) (Revised by Mario Carneiro, 3-Jan-2015.) |
| Ref | Expression |
|---|---|
| fineqv | ⊢ (¬ ω ∈ V ↔ Fin = V) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ssv 4007 | . . . 4 ⊢ Fin ⊆ V | |
| 2 | 1 | a1i 11 | . . 3 ⊢ (¬ ω ∈ V → Fin ⊆ V) |
| 3 | vex 3483 | . . . . . . . 8 ⊢ 𝑎 ∈ V | |
| 4 | fineqvlem 9294 | . . . . . . . 8 ⊢ ((𝑎 ∈ V ∧ ¬ 𝑎 ∈ Fin) → ω ≼ 𝒫 𝒫 𝑎) | |
| 5 | 3, 4 | mpan 690 | . . . . . . 7 ⊢ (¬ 𝑎 ∈ Fin → ω ≼ 𝒫 𝒫 𝑎) |
| 6 | reldom 8987 | . . . . . . . 8 ⊢ Rel ≼ | |
| 7 | 6 | brrelex1i 5739 | . . . . . . 7 ⊢ (ω ≼ 𝒫 𝒫 𝑎 → ω ∈ V) |
| 8 | 5, 7 | syl 17 | . . . . . 6 ⊢ (¬ 𝑎 ∈ Fin → ω ∈ V) |
| 9 | 8 | con1i 147 | . . . . 5 ⊢ (¬ ω ∈ V → 𝑎 ∈ Fin) |
| 10 | 9 | a1d 25 | . . . 4 ⊢ (¬ ω ∈ V → (𝑎 ∈ V → 𝑎 ∈ Fin)) |
| 11 | 10 | ssrdv 3988 | . . 3 ⊢ (¬ ω ∈ V → V ⊆ Fin) |
| 12 | 2, 11 | eqssd 4000 | . 2 ⊢ (¬ ω ∈ V → Fin = V) |
| 13 | ominf 9290 | . . 3 ⊢ ¬ ω ∈ Fin | |
| 14 | eleq2 2829 | . . 3 ⊢ (Fin = V → (ω ∈ Fin ↔ ω ∈ V)) | |
| 15 | 13, 14 | mtbii 326 | . 2 ⊢ (Fin = V → ¬ ω ∈ V) |
| 16 | 12, 15 | impbii 209 | 1 ⊢ (¬ ω ∈ V ↔ Fin = V) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ↔ wb 206 = wceq 1540 ∈ wcel 2108 Vcvv 3479 ⊆ wss 3950 𝒫 cpw 4598 class class class wbr 5141 ωcom 7883 ≼ cdom 8979 Fincfn 8981 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5277 ax-sep 5294 ax-nul 5304 ax-pow 5363 ax-pr 5430 ax-un 7751 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-iun 4991 df-br 5142 df-opab 5204 df-mpt 5224 df-tr 5258 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5635 df-we 5637 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-ord 6385 df-on 6386 df-lim 6387 df-suc 6388 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-om 7884 df-1o 8502 df-er 8741 df-en 8982 df-dom 8983 df-sdom 8984 df-fin 8985 |
| This theorem is referenced by: npomex 11032 finorwe 37376 |
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