Theorem ctex 8973

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 8962	. 2 ⊢ Rel ≼
2	1	brrelex1i 5715	1 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  Vcvv 3453   class class class wbr 5107  ωcom 7866   ≼ cdom 8954

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-pr 5402

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 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-xp 5665  df-rel 5666  df-dom 8958

This theorem is used by:  cnvct  9045  xpct  10023  iunfictbso  10121  unctb  10210  dmct  10530  dmctOLD  10531  fimactOLD  10544  fnct  10548  fnctOLD  10549  mptct  10550  iunctb  10587  cctop  23237  1stcrestlem  23683  2ndcdisj2  23689  dis2ndc  23692  uniiccdif  25812  mptctf  33195  elsigagen2  34667  measvunilem  34731  measvunilem0  34732  measvuni  34733  sxbrsigalem1  34804  omssubadd  34819  carsggect  34837  pmeasadd  34844  mpct  46040  axccdom  46060  rn1st  46110