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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 4019 | . . . 4 ⊢ Fin ⊆ V | |
2 | 1 | a1i 11 | . . 3 ⊢ (¬ ω ∈ V → Fin ⊆ V) |
3 | vex 3481 | . . . . . . . 8 ⊢ 𝑎 ∈ V | |
4 | fineqvlem 9295 | . . . . . . . 8 ⊢ ((𝑎 ∈ V ∧ ¬ 𝑎 ∈ Fin) → ω ≼ 𝒫 𝒫 𝑎) | |
5 | 3, 4 | mpan 690 | . . . . . . 7 ⊢ (¬ 𝑎 ∈ Fin → ω ≼ 𝒫 𝒫 𝑎) |
6 | reldom 8989 | . . . . . . . 8 ⊢ Rel ≼ | |
7 | 6 | brrelex1i 5744 | . . . . . . 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 4000 | . . 3 ⊢ (¬ ω ∈ V → V ⊆ Fin) |
12 | 2, 11 | eqssd 4012 | . 2 ⊢ (¬ ω ∈ V → Fin = V) |
13 | ominf 9291 | . . 3 ⊢ ¬ ω ∈ Fin | |
14 | eleq2 2827 | . . 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 1536 ∈ wcel 2105 Vcvv 3477 ⊆ wss 3962 𝒫 cpw 4604 class class class wbr 5147 ωcom 7886 ≼ cdom 8981 Fincfn 8983 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-om 7887 df-1o 8504 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 |
This theorem is referenced by: npomex 11033 finorwe 37364 |
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