Theorem ctex 8621

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 8610	. 2 ⊢ Rel ≼
2	1	brrelex1i 5590	1 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2112  Vcvv 3398   class class class wbr 5039  ωcom 7622   ≼ cdom 8602

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pr 5307

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3400  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534  df-br 5040  df-opab 5102  df-xp 5542  df-rel 5543  df-dom 8606

This theorem is referenced by:  cnvct  8689  ssct  8704  xpct  9595  iunfictbso  9693  unctb  9784  dmct  10103  fimact  10114  fnct  10116  mptct  10117  iunctb  10153  cctop  21857  1stcrestlem  22303  2ndcdisj2  22308  dis2ndc  22311  uniiccdif  24429  mptctf  30726  elsigagen2  31782  measvunilem  31846  measvunilem0  31847  measvuni  31848  sxbrsigalem1  31918  omssubadd  31933  carsggect  31951  pmeasadd  31958  mpct  42355  axccdom  42376