Theorem 2oex 8405

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 7677. (Proof shortened by Zhi Wang, 19-Sep-2024.)

Assertion

Ref	Expression
2oex	⊢ 2o ∈ V

Proof of Theorem 2oex

Step	Hyp	Ref	Expression
1		df2o3 8402	. 2 ⊢ 2o = {∅, 1o}
2		prex 5379	. 2 ⊢ {∅, 1o} ∈ V
3	1, 2	eqeltri 2829	1 ⊢ 2o ∈ V

Syntax hints:   ∈ wcel 2113  Vcvv 3438  ∅c0 4284  {cpr 4579  1_oc1o 8387  2_oc2o 8388

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 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

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 2712  df-cleq 2725  df-clel 2808  df-v 3440  df-dif 3902  df-un 3904  df-nul 4285  df-sn 4578  df-pr 4580  df-suc 6320  df-1o 8394  df-2o 8395

This theorem is referenced by:  2on  8407  snnen2o  9139  1sdom2  9142  setc2obas  18011  setc2ohom  18012  nogt01o  27645  nosupbday  27654  noetainflem1  27686  noetainflem2  27687  noetainflem4  27689  fmlaomn0  35445  goaln0  35448  goalrlem  35451  goalr  35452  fmlasucdisj  35454  satffunlem1lem1  35457  satffunlem2lem1  35459  ex-sategoelel12  35482  oenord1ex  43422  onno  43540  clsk1indlem1  44152  clsk1independent  44153  nelsubc3  49186  setc2othin  49581  setc1onsubc  49717