Theorem 2oex 8464

Description: 2_o is a set. (Contributed by BJ, 6-Apr-2019.) Remove dependency on ax-10 2182, ax-11 2198, ax-12 2219, ax-un 7733. (Proof shortened by Zhi Wang, 19-Sep-2024.)

Assertion

Ref	Expression
2oex	⊢ 2o ∈ V

Proof of Theorem 2oex

Step	Hyp	Ref	Expression
1		df2o3 8460	. 2 ⊢ 2o = {∅, 1o}
2		prex 5410	. 2 ⊢ {∅, 1o} ∈ V
3	1, 2	eqeltri 2865	1 ⊢ 2o ∈ V

Syntax hints:   ∈ wcel 2149  Vcvv 3463  ∅c0 4294  {cpr 4596  1_oc1o 8445  2_oc2o 8446

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3465  df-dif 3916  df-un 3918  df-nul 4295  df-sn 4595  df-pr 4597  df-suc 6367  df-1o 8452  df-2o 8453

This theorem is referenced by:  2on  8466  snnen2o  9204  1sdom2  9207  setc2obas  18150  setc2ohom  18151  nogt01o  27825  nosupbday  27834  noetainflem1  27866  noetainflem2  27867  noetainflem4  27869  fmlaomn0  35780  goaln0  35783  goalrlem  35786  goalr  35787  fmlasucdisj  35789  satffunlem1lem1  35792  satffunlem2lem1  35794  ex-sategoelel12  35817  oenord1ex  43933  onnoxp  44050  clsk1indlem1  44662  clsk1independent  44663  nelsubc3  49733  setc2othin  50128  setc1onsubc  50264