Theorem ctex 8974

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 8963	. 2 ⊢ Rel ≼
2	1	brrelex1i 5707	1 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  Vcvv 3451   class class class wbr 5103  ωcom 7866   ≼ cdom 8955

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 2733  ax-sep 5249  ax-pr 5391

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 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5657  df-rel 5658  df-dom 8959

This theorem is used by:  cnvct  9046  xpct  10076  iunfictbso  10174  unctb  10263  dmct  10583  dmctOLD  10584  fimactOLD  10597  fnct  10601  fnctOLD  10602  mptct  10603  iunctb  10640  cctop  23304  1stcrestlem  23750  2ndcdisj2  23756  dis2ndc  23759  uniiccdif  25879  mptctf  33290  elsigagen2  34763  measvunilem  34827  measvunilem0  34828  measvuni  34829  sxbrsigalem1  34900  omssubadd  34915  carsggect  34933  pmeasadd  34940  mpct  46158  axccdom  46178  rn1st  46228