Theorem ctex 8935

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 8924	. 2 ⊢ Rel ≼
2	1	brrelex1i 5694	1 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2109  Vcvv 3447   class class class wbr 5107  ωcom 7842   ≼ cdom 8916

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-xp 5644  df-rel 5645  df-dom 8920

This theorem is referenced by:  cnvct  9005  xpct  9969  iunfictbso  10067  unctb  10157  dmct  10477  fimact  10488  fnct  10490  mptct  10491  iunctb  10527  cctop  22893  1stcrestlem  23339  2ndcdisj2  23344  dis2ndc  23347  uniiccdif  25479  mptctf  32641  elsigagen2  34138  measvunilem  34202  measvunilem0  34203  measvuni  34204  sxbrsigalem1  34276  omssubadd  34291  carsggect  34309  pmeasadd  34316  mpct  45195  axccdom  45216  rn1st  45267