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

Theorem ctex 7856
 Description: A countable set is a set. (Contributed by Thierry Arnoux, 29-Dec-2016.)
Assertion
Ref Expression
ctex (𝐴 ≼ ω → 𝐴 ∈ V)

Proof of Theorem ctex
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 brdomi 7852 . 2 (𝐴 ≼ ω → ∃𝑓 𝑓:𝐴1-1→ω)
2 f1dm 6018 . . . 4 (𝑓:𝐴1-1→ω → dom 𝑓 = 𝐴)
3 vex 3176 . . . . 5 𝑓 ∈ V
43dmex 6991 . . . 4 dom 𝑓 ∈ V
52, 4syl6eqelr 2697 . . 3 (𝑓:𝐴1-1→ω → 𝐴 ∈ V)
65exlimiv 1845 . 2 (∃𝑓 𝑓:𝐴1-1→ω → 𝐴 ∈ V)
71, 6syl 17 1 (𝐴 ≼ ω → 𝐴 ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∃wex 1695   ∈ wcel 1977  Vcvv 3173   class class class wbr 4583  dom cdm 5038  –1-1→wf1 5801  ωcom 6957   ≼ cdom 7839 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-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  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-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-dm 5048  df-rn 5049  df-fn 5807  df-f 5808  df-f1 5809  df-dom 7843 This theorem is referenced by:  ssct  7926  xpct  8722  fimact  9238  cctop  20620  fnct  28876  dmct  28877  cnvct  28878  mptct  28880  mptctf  28883  elsigagen2  29538  measvunilem  29602  measvunilem0  29603  measvuni  29604  sxbrsigalem1  29674  omssubadd  29689  carsggect  29707  pmeasadd  29714  mpct  38388  axccdom  38411
 Copyright terms: Public domain W3C validator