Theorem 2oex 8298

Description: 2_o is a set. (Contributed by BJ, 6-Apr-2019.) Remove dependency on ax-10 2140, ax-11 2157, ax-12 2174, ax-un 7579. (Proof shortened by Zhi Wang, 19-Sep-2024.)

Assertion

Ref	Expression
2oex	⊢ 2o ∈ V

Proof of Theorem 2oex

Step	Hyp	Ref	Expression
1		df2o3 8296	. 2 ⊢ 2o = {∅, 1o}
2		prex 5358	. 2 ⊢ {∅, 1o} ∈ V
3	1, 2	eqeltri 2836	1 ⊢ 2o ∈ V

Syntax hints:   ∈ wcel 2109  Vcvv 3430  ∅c0 4261  {cpr 4568  1_oc1o 8274  2_oc2o 8275

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pr 5355

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1544  df-fal 1554  df-ex 1786  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-v 3432  df-dif 3894  df-un 3896  df-nul 4262  df-sn 4567  df-pr 4569  df-suc 6269  df-1o 8281  df-2o 8282

This theorem is referenced by:  setc2obas  17790  setc2ohom  17791  fmlaomn0  33331  goaln0  33334  goalrlem  33337  goalr  33338  fmlasucdisj  33340  satffunlem1lem1  33343  satffunlem2lem1  33345  ex-sategoelel12  33368  nogt01o  33878  nosupbday  33887  noetainflem1  33919  noetainflem2  33920  noetainflem4  33922  clsk1indlem1  41608  clsk1independent  41609  setc2othin  46289