Theorem ctex 8993

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 8982	. 2 ⊢ Rel ≼
2	1	brrelex1i 5706	1 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2144  Vcvv 3450   class class class wbr 5102  ωcom 7877   ≼ cdom 8974

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-xp 5656  df-rel 5657  df-dom 8978

This theorem is used by:  cnvct  9065  xpct  10098  iunfictbso  10196  unctb  10285  dmct  10605  dmctOLD  10606  fimactOLD  10619  fnct  10623  fnctOLD  10624  mptct  10625  iunctb  10662  cctop  23329  1stcrestlem  23775  2ndcdisj2  23781  dis2ndc  23784  uniiccdif  25904  mptctf  33355  elsigagen2  34827  measvunilem  34891  measvunilem0  34892  measvuni  34893  sxbrsigalem1  34963  omssubadd  34978  carsggect  34996  pmeasadd  35003  mpct  46308  axccdom  46328  rn1st  46378