Theorem ctex 8910

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 8899	. 2 ⊢ Rel ≼
2	1	brrelex1i 5687	1 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2114  Vcvv 3429   class class class wbr 5085  ωcom 7817   ≼ cdom 8891

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-xp 5637  df-rel 5638  df-dom 8895

This theorem is referenced by:  cnvct  8981  xpct  9938  iunfictbso  10036  unctb  10126  dmct  10446  fimact  10457  fnct  10459  mptct  10460  iunctb  10497  cctop  22971  1stcrestlem  23417  2ndcdisj2  23422  dis2ndc  23425  uniiccdif  25545  mptctf  32789  elsigagen2  34292  measvunilem  34356  measvunilem0  34357  measvuni  34358  sxbrsigalem1  34429  omssubadd  34444  carsggect  34462  pmeasadd  34469  mpct  45630  axccdom  45651  rn1st  45702