Theorem 2oex 8517

Description: 2_o is a set. (Contributed by BJ, 6-Apr-2019.) Remove dependency on ax-10 2141, ax-11 2157, ax-12 2177, ax-un 7755. (Proof shortened by Zhi Wang, 19-Sep-2024.)

Assertion

Ref	Expression
2oex	⊢ 2o ∈ V

Proof of Theorem 2oex

Step	Hyp	Ref	Expression
1		df2o3 8514	. 2 ⊢ 2o = {∅, 1o}
2		prex 5437	. 2 ⊢ {∅, 1o} ∈ V
3	1, 2	eqeltri 2837	1 ⊢ 2o ∈ V

Syntax hints:   ∈ wcel 2108  Vcvv 3480  ∅c0 4333  {cpr 4628  1_oc1o 8499  2_oc2o 8500

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 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-dif 3954  df-un 3956  df-nul 4334  df-sn 4627  df-pr 4629  df-suc 6390  df-1o 8506  df-2o 8507

This theorem is referenced by:  2on  8520  snnen2o  9273  1sdom2  9276  setc2obas  18139  setc2ohom  18140  nogt01o  27741  nosupbday  27750  noetainflem1  27782  noetainflem2  27783  noetainflem4  27785  fmlaomn0  35395  goaln0  35398  goalrlem  35401  goalr  35402  fmlasucdisj  35404  satffunlem1lem1  35407  satffunlem2lem1  35409  ex-sategoelel12  35432  oenord1ex  43328  onno  43446  clsk1indlem1  44058  clsk1independent  44059  setc2othin  49113