Theorem ctex 8900

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 8889	. 2 ⊢ Rel ≼
2	1	brrelex1i 5680	1 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2113  Vcvv 3440   class class class wbr 5098  ωcom 7808   ≼ cdom 8881

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 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

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 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-xp 5630  df-rel 5631  df-dom 8885

This theorem is referenced by:  cnvct  8971  xpct  9926  iunfictbso  10024  unctb  10114  dmct  10434  fimact  10445  fnct  10447  mptct  10448  iunctb  10485  cctop  22950  1stcrestlem  23396  2ndcdisj2  23401  dis2ndc  23404  uniiccdif  25535  mptctf  32795  elsigagen2  34305  measvunilem  34369  measvunilem0  34370  measvuni  34371  sxbrsigalem1  34442  omssubadd  34457  carsggect  34475  pmeasadd  34482  mpct  45441  axccdom  45462  rn1st  45513