Theorem ctex 8896

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 8885	. 2 ⊢ Rel ≼
2	1	brrelex1i 5679	1 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2109  Vcvv 3438   class class class wbr 5095  ωcom 7806   ≼ cdom 8877

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

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 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5096  df-opab 5158  df-xp 5629  df-rel 5630  df-dom 8881

This theorem is referenced by:  cnvct  8966  xpct  9929  iunfictbso  10027  unctb  10117  dmct  10437  fimact  10448  fnct  10450  mptct  10451  iunctb  10487  cctop  22909  1stcrestlem  23355  2ndcdisj2  23360  dis2ndc  23363  uniiccdif  25495  mptctf  32674  elsigagen2  34114  measvunilem  34178  measvunilem0  34179  measvuni  34180  sxbrsigalem1  34252  omssubadd  34267  carsggect  34285  pmeasadd  34292  mpct  45179  axccdom  45200  rn1st  45251