Theorem ctex 8907

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 8896	. 2 ⊢ Rel ≼
2	1	brrelex1i 5681	1 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2119  Vcvv 3432   class class class wbr 5079  ωcom 7813   ≼ cdom 8888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-br 5080  df-opab 5142  df-xp 5631  df-rel 5632  df-dom 8892

This theorem is referenced by:  cnvct  8978  xpct  9936  iunfictbso  10034  unctb  10124  dmct  10444  fimact  10455  fnct  10457  mptct  10458  iunctb  10495  cctop  22996  1stcrestlem  23442  2ndcdisj2  23447  dis2ndc  23450  uniiccdif  25570  mptctf  32815  elsigagen2  34339  measvunilem  34403  measvunilem0  34404  measvuni  34405  sxbrsigalem1  34476  omssubadd  34491  carsggect  34509  pmeasadd  34516  mpct  45654  axccdom  45674  rn1st  45724