![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 4006 | . . . 4 ⊢ Fin ⊆ V | |
2 | 1 | a1i 11 | . . 3 ⊢ (¬ ω ∈ V → Fin ⊆ V) |
3 | vex 3477 | . . . . . . . 8 ⊢ 𝑎 ∈ V | |
4 | fineqvlem 9293 | . . . . . . . 8 ⊢ ((𝑎 ∈ V ∧ ¬ 𝑎 ∈ Fin) → ω ≼ 𝒫 𝒫 𝑎) | |
5 | 3, 4 | mpan 688 | . . . . . . 7 ⊢ (¬ 𝑎 ∈ Fin → ω ≼ 𝒫 𝒫 𝑎) |
6 | reldom 8976 | . . . . . . . 8 ⊢ Rel ≼ | |
7 | 6 | brrelex1i 5738 | . . . . . . 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 3999 | . 2 ⊢ (¬ ω ∈ V → Fin = V) |
13 | ominf 9289 | . . 3 ⊢ ¬ ω ∈ Fin | |
14 | eleq2 2818 | . . 3 ⊢ (Fin = V → (ω ∈ Fin ↔ ω ∈ V)) | |
15 | 13, 14 | mtbii 325 | . 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 1533 ∈ wcel 2098 Vcvv 3473 ⊆ wss 3949 𝒫 cpw 4606 class class class wbr 5152 ωcom 7876 ≼ cdom 8968 Fincfn 8970 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-om 7877 df-1o 8493 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 |
This theorem is referenced by: npomex 11027 finorwe 36894 |
Copyright terms: Public domain | W3C validator |