Theorem 2oex 8406

Description: 2_o is a set. (Contributed by BJ, 6-Apr-2019.) Remove dependency on ax-10 2146, ax-11 2162, ax-12 2182, ax-un 7678. (Proof shortened by Zhi Wang, 19-Sep-2024.)

Assertion

Ref	Expression
2oex	⊢ 2o ∈ V

Proof of Theorem 2oex

Step	Hyp	Ref	Expression
1		df2o3 8403	. 2 ⊢ 2o = {∅, 1o}
2		prex 5380	. 2 ⊢ {∅, 1o} ∈ V
3	1, 2	eqeltri 2830	1 ⊢ 2o ∈ V

Syntax hints:   ∈ wcel 2113  Vcvv 3438  ∅c0 4283  {cpr 4580  1_oc1o 8388  2_oc2o 8389

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 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-v 3440  df-dif 3902  df-un 3904  df-nul 4284  df-sn 4579  df-pr 4581  df-suc 6321  df-1o 8395  df-2o 8396

This theorem is referenced by:  2on  8408  snnen2o  9143  1sdom2  9146  setc2obas  18016  setc2ohom  18017  nogt01o  27662  nosupbday  27671  noetainflem1  27703  noetainflem2  27704  noetainflem4  27706  fmlaomn0  35533  goaln0  35536  goalrlem  35539  goalr  35540  fmlasucdisj  35542  satffunlem1lem1  35545  satffunlem2lem1  35547  ex-sategoelel12  35570  oenord1ex  43499  onno  43616  clsk1indlem1  44228  clsk1independent  44229  nelsubc3  49258  setc2othin  49653  setc1onsubc  49789