Theorem 2oex 8445

Description: 2_o is a set. (Contributed by BJ, 6-Apr-2019.) Remove dependency on ax-10 2142, ax-11 2158, ax-12 2178, ax-un 7711. (Proof shortened by Zhi Wang, 19-Sep-2024.)

Assertion

Ref	Expression
2oex	⊢ 2o ∈ V

Proof of Theorem 2oex

Step	Hyp	Ref	Expression
1		df2o3 8442	. 2 ⊢ 2o = {∅, 1o}
2		prex 5392	. 2 ⊢ {∅, 1o} ∈ V
3	1, 2	eqeltri 2824	1 ⊢ 2o ∈ V

Syntax hints:   ∈ wcel 2109  Vcvv 3447  ∅c0 4296  {cpr 4591  1_oc1o 8427  2_oc2o 8428

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3449  df-dif 3917  df-un 3919  df-nul 4297  df-sn 4590  df-pr 4592  df-suc 6338  df-1o 8434  df-2o 8435

This theorem is referenced by:  2on  8447  snnen2o  9184  1sdom2  9187  setc2obas  18056  setc2ohom  18057  nogt01o  27608  nosupbday  27617  noetainflem1  27649  noetainflem2  27650  noetainflem4  27652  fmlaomn0  35377  goaln0  35380  goalrlem  35383  goalr  35384  fmlasucdisj  35386  satffunlem1lem1  35389  satffunlem2lem1  35391  ex-sategoelel12  35414  oenord1ex  43304  onno  43422  clsk1indlem1  44034  clsk1independent  44035  nelsubc3  49060  setc2othin  49455  setc1onsubc  49591