Theorem ctex 9021

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 9010	. 2 ⊢ Rel ≼
2	1	brrelex1i 5712	1 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147  Vcvv 3454   class class class wbr 5106  ωcom 7904   ≼ cdom 9002

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5253  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5172  df-xp 5662  df-rel 5663  df-dom 9006

This theorem is used by:  cnvct  9093  xpct  10125  iunfictbso  10223  unctb  10312  dmct  10632  dmctOLD  10633  fimactOLD  10646  fnct  10650  fnctOLD  10651  mptct  10652  iunctb  10689  cctop  23357  1stcrestlem  23803  2ndcdisj2  23809  dis2ndc  23812  uniiccdif  25932  mptctf  33382  elsigagen2  34864  measvunilem  34928  measvunilem0  34929  measvuni  34930  sxbrsigalem1  35000  omssubadd  35015  carsggect  35033  pmeasadd  35040  mpct  46430  axccdom  46450  rn1st  46500