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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 3874 | . . . 4 ⊢ Fin ⊆ V | |
2 | 1 | a1i 11 | . . 3 ⊢ (¬ ω ∈ V → Fin ⊆ V) |
3 | vex 3411 | . . . . . . . 8 ⊢ 𝑎 ∈ V | |
4 | fineqvlem 8525 | . . . . . . . 8 ⊢ ((𝑎 ∈ V ∧ ¬ 𝑎 ∈ Fin) → ω ≼ 𝒫 𝒫 𝑎) | |
5 | 3, 4 | mpan 678 | . . . . . . 7 ⊢ (¬ 𝑎 ∈ Fin → ω ≼ 𝒫 𝒫 𝑎) |
6 | reldom 8310 | . . . . . . . 8 ⊢ Rel ≼ | |
7 | 6 | brrelex1i 5454 | . . . . . . 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 3857 | . . 3 ⊢ (¬ ω ∈ V → V ⊆ Fin) |
12 | 2, 11 | eqssd 3868 | . 2 ⊢ (¬ ω ∈ V → Fin = V) |
13 | ominf 8523 | . . 3 ⊢ ¬ ω ∈ Fin | |
14 | eleq2 2847 | . . 3 ⊢ (Fin = V → (ω ∈ Fin ↔ ω ∈ V)) | |
15 | 13, 14 | mtbii 318 | . 2 ⊢ (Fin = V → ¬ ω ∈ V) |
16 | 12, 15 | impbii 201 | 1 ⊢ (¬ ω ∈ V ↔ Fin = V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 198 = wceq 1508 ∈ wcel 2051 Vcvv 3408 ⊆ wss 3822 𝒫 cpw 4416 class class class wbr 4925 ωcom 7394 ≼ cdom 8302 Fincfn 8304 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-ral 3086 df-rex 3087 df-reu 3088 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-om 7395 df-er 8087 df-en 8305 df-dom 8306 df-sdom 8307 df-fin 8308 |
This theorem is referenced by: npomex 10214 |
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