Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  fineqv Structured version   Visualization version   GIF version

Theorem fineqv 8060
 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.)
Assertion
Ref Expression
fineqv (¬ ω ∈ V ↔ Fin = V)

Proof of Theorem fineqv
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 ssv 3588 . . . 4 Fin ⊆ V
21a1i 11 . . 3 (¬ ω ∈ V → Fin ⊆ V)
3 vex 3176 . . . . . . . 8 𝑎 ∈ V
4 fineqvlem 8059 . . . . . . . 8 ((𝑎 ∈ V ∧ ¬ 𝑎 ∈ Fin) → ω ≼ 𝒫 𝒫 𝑎)
53, 4mpan 702 . . . . . . 7 𝑎 ∈ Fin → ω ≼ 𝒫 𝒫 𝑎)
6 reldom 7847 . . . . . . . 8 Rel ≼
76brrelexi 5082 . . . . . . 7 (ω ≼ 𝒫 𝒫 𝑎 → ω ∈ V)
85, 7syl 17 . . . . . 6 𝑎 ∈ Fin → ω ∈ V)
98con1i 143 . . . . 5 (¬ ω ∈ V → 𝑎 ∈ Fin)
109a1d 25 . . . 4 (¬ ω ∈ V → (𝑎 ∈ V → 𝑎 ∈ Fin))
1110ssrdv 3574 . . 3 (¬ ω ∈ V → V ⊆ Fin)
122, 11eqssd 3585 . 2 (¬ ω ∈ V → Fin = V)
13 ominf 8057 . . 3 ¬ ω ∈ Fin
14 eleq2 2677 . . 3 (Fin = V → (ω ∈ Fin ↔ ω ∈ V))
1513, 14mtbii 315 . 2 (Fin = V → ¬ ω ∈ V)
1612, 15impbii 198 1 (¬ ω ∈ V ↔ Fin = V)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 195   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540  𝒫 cpw 4108   class class class wbr 4583  ωcom 6957   ≼ cdom 7839  Fincfn 7841 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845 This theorem is referenced by:  npomex  9697
 Copyright terms: Public domain W3C validator