Theorem ctex 8959

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 8945	. 2 ⊢ Rel ≼
2	1	brrelex1i 5733	1 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2107  Vcvv 3475   class class class wbr 5149  ωcom 7855   ≼ cdom 8937

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-dom 8941

This theorem is referenced by:  cnvct  9034  ssctOLD  9052  xpct  10011  iunfictbso  10109  unctb  10200  dmct  10519  fimact  10530  fnct  10532  mptct  10533  iunctb  10569  cctop  22509  1stcrestlem  22956  2ndcdisj2  22961  dis2ndc  22964  uniiccdif  25095  mptctf  31942  elsigagen2  33146  measvunilem  33210  measvunilem0  33211  measvuni  33212  sxbrsigalem1  33284  omssubadd  33299  carsggect  33317  pmeasadd  33324  mpct  43900  axccdom  43921  rn1st  43978