Theorem ctex 8708

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 8697	. 2 ⊢ Rel ≼
2	1	brrelex1i 5634	1 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2108  Vcvv 3422   class class class wbr 5070  ωcom 7687   ≼ cdom 8689

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-xp 5586  df-rel 5587  df-dom 8693

This theorem is referenced by:  cnvct  8778  ssct  8793  xpct  9703  iunfictbso  9801  unctb  9892  dmct  10211  fimact  10222  fnct  10224  mptct  10225  iunctb  10261  cctop  22064  1stcrestlem  22511  2ndcdisj2  22516  dis2ndc  22519  uniiccdif  24647  mptctf  30954  elsigagen2  32016  measvunilem  32080  measvunilem0  32081  measvuni  32082  sxbrsigalem1  32152  omssubadd  32167  carsggect  32185  pmeasadd  32192  mpct  42630  axccdom  42651