Theorem ctex 8903

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 8892	. 2 ⊢ Rel ≼
2	1	brrelex1i 5680	1 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2114  Vcvv 3430   class class class wbr 5086  ωcom 7810   ≼ cdom 8884

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 2709  ax-sep 5231  ax-pr 5370

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 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-xp 5630  df-rel 5631  df-dom 8888

This theorem is referenced by:  cnvct  8974  xpct  9929  iunfictbso  10027  unctb  10117  dmct  10437  fimact  10448  fnct  10450  mptct  10451  iunctb  10488  cctop  22981  1stcrestlem  23427  2ndcdisj2  23432  dis2ndc  23435  uniiccdif  25555  mptctf  32804  elsigagen2  34308  measvunilem  34372  measvunilem0  34373  measvuni  34374  sxbrsigalem1  34445  omssubadd  34460  carsggect  34478  pmeasadd  34485  mpct  45648  axccdom  45669  rn1st  45720