Theorem 2oex 8339

Description: 2_o is a set. (Contributed by BJ, 6-Apr-2019.) Remove dependency on ax-10 2135, ax-11 2152, ax-12 2169, ax-un 7620. (Proof shortened by Zhi Wang, 19-Sep-2024.)

Assertion

Ref	Expression
2oex	⊢ 2o ∈ V

Proof of Theorem 2oex

Step	Hyp	Ref	Expression
1		df2o3 8336	. 2 ⊢ 2o = {∅, 1o}
2		prex 5364	. 2 ⊢ {∅, 1o} ∈ V
3	1, 2	eqeltri 2833	1 ⊢ 2o ∈ V

Syntax hints:   ∈ wcel 2104  Vcvv 3437  ∅c0 4262  {cpr 4567  1_oc1o 8321  2_oc2o 8322

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pr 5361

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3439  df-dif 3895  df-un 3897  df-nul 4263  df-sn 4566  df-pr 4568  df-suc 6287  df-1o 8328  df-2o 8329

This theorem is referenced by:  2on  8342  snnen2o  9058  1sdom2  9061  setc2obas  17854  setc2ohom  17855  fmlaomn0  33397  goaln0  33400  goalrlem  33403  goalr  33404  fmlasucdisj  33406  satffunlem1lem1  33409  satffunlem2lem1  33411  ex-sategoelel12  33434  nogt01o  33944  nosupbday  33953  noetainflem1  33985  noetainflem2  33986  noetainflem4  33988  clsk1indlem1  41693  clsk1independent  41694  setc2othin  46395