Theorem ctex 8940

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 8929	. 2 ⊢ Rel ≼
2	1	brrelex1i 5701	1 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2141  Vcvv 3453   class class class wbr 5099  ωcom 7842   ≼ cdom 8921

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5245  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-xp 5651  df-rel 5652  df-dom 8925

This theorem is referenced by:  cnvct  9011  xpct  9969  iunfictbso  10067  unctb  10157  dmct  10478  fimact  10489  fnct  10491  mptct  10492  iunctb  10529  cctop  23046  1stcrestlem  23492  2ndcdisj2  23497  dis2ndc  23500  uniiccdif  25620  mptctf  32868  elsigagen2  34406  measvunilem  34470  measvunilem0  34471  measvuni  34472  sxbrsigalem1  34543  omssubadd  34558  carsggect  34576  pmeasadd  34583  mpct  45742  axccdom  45762  rn1st  45812