Theorem 2oex 8477

Description: 2_o is a set. (Contributed by BJ, 6-Apr-2019.) Remove dependency on ax-10 2138, ax-11 2155, ax-12 2172, ax-un 7725. (Proof shortened by Zhi Wang, 19-Sep-2024.)

Assertion

Ref	Expression
2oex	⊢ 2o ∈ V

Proof of Theorem 2oex

Step	Hyp	Ref	Expression
1		df2o3 8474	. 2 ⊢ 2o = {∅, 1o}
2		prex 5433	. 2 ⊢ {∅, 1o} ∈ V
3	1, 2	eqeltri 2830	1 ⊢ 2o ∈ V

Syntax hints:   ∈ wcel 2107  Vcvv 3475  ∅c0 4323  {cpr 4631  1_oc1o 8459  2_oc2o 8460

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-dif 3952  df-un 3954  df-nul 4324  df-sn 4630  df-pr 4632  df-suc 6371  df-1o 8466  df-2o 8467

This theorem is referenced by:  2on  8480  snnen2o  9237  1sdom2  9240  setc2obas  18044  setc2ohom  18045  nogt01o  27199  nosupbday  27208  noetainflem1  27240  noetainflem2  27241  noetainflem4  27243  fmlaomn0  34381  goaln0  34384  goalrlem  34387  goalr  34388  fmlasucdisj  34390  satffunlem1lem1  34393  satffunlem2lem1  34395  ex-sategoelel12  34418  oenord1ex  42065  onno  42184  clsk1indlem1  42796  clsk1independent  42797  setc2othin  47676