Theorem ctex 9023

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 9009	. 2 ⊢ Rel ≼
2	1	brrelex1i 5756	1 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2108  Vcvv 3488   class class class wbr 5166  ωcom 7903   ≼ cdom 9001

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-rel 5707  df-dom 9005

This theorem is referenced by:  cnvct  9099  ssctOLD  9118  xpct  10085  iunfictbso  10183  unctb  10273  dmct  10593  fimact  10604  fnct  10606  mptct  10607  iunctb  10643  cctop  23034  1stcrestlem  23481  2ndcdisj2  23486  dis2ndc  23489  uniiccdif  25632  mptctf  32731  elsigagen2  34112  measvunilem  34176  measvunilem0  34177  measvuni  34178  sxbrsigalem1  34250  omssubadd  34265  carsggect  34283  pmeasadd  34290  mpct  45108  axccdom  45129  rn1st  45183