Theorem ctex 9004

Description: A countable set is a set. (Contributed by Thierry Arnoux, 29-Dec-2016.) (Proof shortened by Jim Kingdon, 13-Mar-2023.)

Assertion

Ref	Expression
ctex	⊢ (𝐴 ≼ ω → 𝐴 ∈ V)

Proof of Theorem ctex

Step	Hyp	Ref	Expression
1		reldom 8991	. 2 ⊢ Rel ≼
2	1	brrelex1i 5741	1 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2108  Vcvv 3480   class class class wbr 5143  ωcom 7887   ≼ cdom 8983

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-dom 8987

This theorem is referenced by:  cnvct  9074  ssctOLD  9092  xpct  10056  iunfictbso  10154  unctb  10244  dmct  10564  fimact  10575  fnct  10577  mptct  10578  iunctb  10614  cctop  23013  1stcrestlem  23460  2ndcdisj2  23465  dis2ndc  23468  uniiccdif  25613  mptctf  32729  elsigagen2  34149  measvunilem  34213  measvunilem0  34214  measvuni  34215  sxbrsigalem1  34287  omssubadd  34302  carsggect  34320  pmeasadd  34327  mpct  45206  axccdom  45227  rn1st  45280