Theorem ctex 8753

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 8739	. 2 ⊢ Rel ≼
2	1	brrelex1i 5643	1 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2106  Vcvv 3432   class class class wbr 5074  ωcom 7712   ≼ cdom 8731

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-xp 5595  df-rel 5596  df-dom 8735

This theorem is referenced by:  cnvct  8824  ssct  8839  xpct  9772  iunfictbso  9870  unctb  9961  dmct  10280  fimact  10291  fnct  10293  mptct  10294  iunctb  10330  cctop  22156  1stcrestlem  22603  2ndcdisj2  22608  dis2ndc  22611  uniiccdif  24742  mptctf  31052  elsigagen2  32116  measvunilem  32180  measvunilem0  32181  measvuni  32182  sxbrsigalem1  32252  omssubadd  32267  carsggect  32285  pmeasadd  32292  mpct  42741  axccdom  42762