Theorem 2oex 8408

Description: 2_o is a set. (Contributed by BJ, 6-Apr-2019.) Remove dependency on ax-10 2146, ax-11 2162, ax-12 2184, ax-un 7680. (Proof shortened by Zhi Wang, 19-Sep-2024.)

Assertion

Ref	Expression
2oex	⊢ 2o ∈ V

Proof of Theorem 2oex

Step	Hyp	Ref	Expression
1		df2o3 8405	. 2 ⊢ 2o = {∅, 1o}
2		prex 5382	. 2 ⊢ {∅, 1o} ∈ V
3	1, 2	eqeltri 2832	1 ⊢ 2o ∈ V

Syntax hints:   ∈ wcel 2113  Vcvv 3440  ∅c0 4285  {cpr 4582  1_oc1o 8390  2_oc2o 8391

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 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

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 2715  df-cleq 2728  df-clel 2811  df-v 3442  df-dif 3904  df-un 3906  df-nul 4286  df-sn 4581  df-pr 4583  df-suc 6323  df-1o 8397  df-2o 8398

This theorem is referenced by:  2on  8410  snnen2o  9145  1sdom2  9148  setc2obas  18018  setc2ohom  18019  nogt01o  27664  nosupbday  27673  noetainflem1  27705  noetainflem2  27706  noetainflem4  27708  fmlaomn0  35584  goaln0  35587  goalrlem  35590  goalr  35591  fmlasucdisj  35593  satffunlem1lem1  35596  satffunlem2lem1  35598  ex-sategoelel12  35621  oenord1ex  43557  onnoxp  43674  clsk1indlem1  44286  clsk1independent  44287  nelsubc3  49316  setc2othin  49711  setc1onsubc  49847