Theorem ctex 8886

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 8875	. 2 ⊢ Rel ≼
2	1	brrelex1i 5672	1 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2111  Vcvv 3436   class class class wbr 5091  ωcom 7796   ≼ cdom 8867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-br 5092  df-opab 5154  df-xp 5622  df-rel 5623  df-dom 8871

This theorem is referenced by:  cnvct  8956  xpct  9904  iunfictbso  10002  unctb  10092  dmct  10412  fimact  10423  fnct  10425  mptct  10426  iunctb  10462  cctop  22919  1stcrestlem  23365  2ndcdisj2  23370  dis2ndc  23373  uniiccdif  25504  mptctf  32694  elsigagen2  34156  measvunilem  34220  measvunilem0  34221  measvuni  34222  sxbrsigalem1  34293  omssubadd  34308  carsggect  34326  pmeasadd  34333  mpct  45237  axccdom  45258  rn1st  45309