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Theorem ctex 7914
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 7910 . 2 (𝐴 ≼ ω → ∃𝑓 𝑓:𝐴1-1→ω)
2 f1dm 6062 . . . 4 (𝑓:𝐴1-1→ω → dom 𝑓 = 𝐴)
3 vex 3189 . . . . 5 𝑓 ∈ V
43dmex 7046 . . . 4 dom 𝑓 ∈ V
52, 4syl6eqelr 2707 . . 3 (𝑓:𝐴1-1→ω → 𝐴 ∈ V)
65exlimiv 1855 . 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 1701  wcel 1987  Vcvv 3186   class class class wbr 4613  dom cdm 5074  1-1wf1 5844  ωcom 7012  cdom 7897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-xp 5080  df-rel 5081  df-cnv 5082  df-dm 5084  df-rn 5085  df-fn 5850  df-f 5851  df-f1 5852  df-dom 7901
This theorem is referenced by:  cnvct  7977  ssct  7985  xpct  8783  dmct  9290  fimact  9301  fnct  9303  mptct  9304  cctop  20720  mptctf  29338  elsigagen2  29992  measvunilem  30056  measvunilem0  30057  measvuni  30058  sxbrsigalem1  30128  omssubadd  30143  carsggect  30161  pmeasadd  30168  mpct  38867  axccdom  38890
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