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Theorem ctex 8136
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 8132 . 2 (𝐴 ≼ ω → ∃𝑓 𝑓:𝐴1-1→ω)
2 f1dm 6266 . . . 4 (𝑓:𝐴1-1→ω → dom 𝑓 = 𝐴)
3 vex 3343 . . . . 5 𝑓 ∈ V
43dmex 7264 . . . 4 dom 𝑓 ∈ V
52, 4syl6eqelr 2848 . . 3 (𝑓:𝐴1-1→ω → 𝐴 ∈ V)
65exlimiv 2007 . 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 1853  wcel 2139  Vcvv 3340   class class class wbr 4804  dom cdm 5266  1-1wf1 6046  ωcom 7230  cdom 8119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-cnv 5274  df-dm 5276  df-rn 5277  df-fn 6052  df-f 6053  df-f1 6054  df-dom 8123
This theorem is referenced by:  cnvct  8198  ssct  8206  xpct  9029  dmct  9538  fimact  9549  fnct  9551  mptct  9552  cctop  21012  mptctf  29804  elsigagen2  30520  measvunilem  30584  measvunilem0  30585  measvuni  30586  sxbrsigalem1  30656  omssubadd  30671  carsggect  30689  pmeasadd  30696  mpct  39892  axccdom  39915
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