Theorem ctex 8961

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 8950	. 2 ⊢ Rel ≼
2	1	brrelex1i 5719	1 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2143  Vcvv 3455   class class class wbr 5110  ωcom 7863   ≼ cdom 8942

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-br 5111  df-opab 5175  df-xp 5669  df-rel 5670  df-dom 8946

This theorem is referenced by:  cnvct  9032  xpct  10001  iunfictbso  10099  unctb  10188  dmct  10509  fimact  10520  fnct  10522  mptct  10523  iunctb  10560  cctop  23144  1stcrestlem  23590  2ndcdisj2  23595  dis2ndc  23598  uniiccdif  25718  mptctf  33039  elsigagen2  34516  measvunilem  34580  measvunilem0  34581  measvuni  34582  sxbrsigalem1  34653  omssubadd  34668  carsggect  34686  pmeasadd  34693  mpct  45898  axccdom  45918  rn1st  45968