Theorem ctex 8892

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 8881	. 2 ⊢ Rel ≼
2	1	brrelex1i 5675	1 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2113  Vcvv 3437   class class class wbr 5093  ωcom 7802   ≼ cdom 8873

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-br 5094  df-opab 5156  df-xp 5625  df-rel 5626  df-dom 8877

This theorem is referenced by:  cnvct  8963  xpct  9914  iunfictbso  10012  unctb  10102  dmct  10422  fimact  10433  fnct  10435  mptct  10436  iunctb  10472  cctop  22922  1stcrestlem  23368  2ndcdisj2  23373  dis2ndc  23376  uniiccdif  25507  mptctf  32703  elsigagen2  34182  measvunilem  34246  measvunilem0  34247  measvuni  34248  sxbrsigalem1  34319  omssubadd  34334  carsggect  34352  pmeasadd  34359  mpct  45322  axccdom  45343  rn1st  45394