Theorem ctex 8978

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 8965	. 2 ⊢ Rel ≼
2	1	brrelex1i 5710	1 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2108  Vcvv 3459   class class class wbr 5119  ωcom 7861   ≼ cdom 8957

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-br 5120  df-opab 5182  df-xp 5660  df-rel 5661  df-dom 8961

This theorem is referenced by:  cnvct  9048  ssctOLD  9066  xpct  10030  iunfictbso  10128  unctb  10218  dmct  10538  fimact  10549  fnct  10551  mptct  10552  iunctb  10588  cctop  22944  1stcrestlem  23390  2ndcdisj2  23395  dis2ndc  23398  uniiccdif  25531  mptctf  32695  elsigagen2  34179  measvunilem  34243  measvunilem0  34244  measvuni  34245  sxbrsigalem1  34317  omssubadd  34332  carsggect  34350  pmeasadd  34357  mpct  45225  axccdom  45246  rn1st  45297