Theorem ctex 9003

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 8990	. 2 ⊢ Rel ≼
2	1	brrelex1i 5745	1 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2106  Vcvv 3478   class class class wbr 5148  ωcom 7887   ≼ cdom 8982

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-dom 8986

This theorem is referenced by:  cnvct  9073  ssctOLD  9091  xpct  10054  iunfictbso  10152  unctb  10242  dmct  10562  fimact  10573  fnct  10575  mptct  10576  iunctb  10612  cctop  23029  1stcrestlem  23476  2ndcdisj2  23481  dis2ndc  23484  uniiccdif  25627  mptctf  32735  elsigagen2  34129  measvunilem  34193  measvunilem0  34194  measvuni  34195  sxbrsigalem1  34267  omssubadd  34282  carsggect  34300  pmeasadd  34307  mpct  45144  axccdom  45165  rn1st  45219