Theorem ctex 8948

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 8937	. 2 ⊢ Rel ≼
2	1	brrelex1i 5708	1 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2145  Vcvv 3457   class class class wbr 5105  ωcom 7850   ≼ cdom 8929

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5106  df-opab 5168  df-xp 5658  df-rel 5659  df-dom 8933

This theorem is referenced by:  cnvct  9019  xpct  9988  iunfictbso  10086  unctb  10175  dmct  10496  fimact  10507  fnct  10509  mptct  10510  iunctb  10547  cctop  23124  1stcrestlem  23570  2ndcdisj2  23575  dis2ndc  23578  uniiccdif  25698  mptctf  32973  elsigagen2  34455  measvunilem  34519  measvunilem0  34520  measvuni  34521  sxbrsigalem1  34592  omssubadd  34607  carsggect  34625  pmeasadd  34632  mpct  45776  axccdom  45796  rn1st  45846