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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 3973 | . . . 4 ⊢ Fin ⊆ V | |
2 | 1 | a1i 11 | . . 3 ⊢ (¬ ω ∈ V → Fin ⊆ V) |
3 | vex 3452 | . . . . . . . 8 ⊢ 𝑎 ∈ V | |
4 | fineqvlem 9213 | . . . . . . . 8 ⊢ ((𝑎 ∈ V ∧ ¬ 𝑎 ∈ Fin) → ω ≼ 𝒫 𝒫 𝑎) | |
5 | 3, 4 | mpan 689 | . . . . . . 7 ⊢ (¬ 𝑎 ∈ Fin → ω ≼ 𝒫 𝒫 𝑎) |
6 | reldom 8896 | . . . . . . . 8 ⊢ Rel ≼ | |
7 | 6 | brrelex1i 5693 | . . . . . . 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 3955 | . . 3 ⊢ (¬ ω ∈ V → V ⊆ Fin) |
12 | 2, 11 | eqssd 3966 | . 2 ⊢ (¬ ω ∈ V → Fin = V) |
13 | ominf 9209 | . . 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 208 | 1 ⊢ (¬ ω ∈ V ↔ Fin = V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 = wceq 1542 ∈ wcel 2107 Vcvv 3448 ⊆ wss 3915 𝒫 cpw 4565 class class class wbr 5110 ωcom 7807 ≼ cdom 8888 Fincfn 8890 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-om 7808 df-1o 8417 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 |
This theorem is referenced by: npomex 10939 finorwe 35882 |
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