Theorem 2oex 8476

Description: 2_o is a set. (Contributed by BJ, 6-Apr-2019.) Remove dependency on ax-10 2137, ax-11 2154, ax-12 2171, ax-un 7724. (Proof shortened by Zhi Wang, 19-Sep-2024.)

Assertion

Ref	Expression
2oex	⊢ 2o ∈ V

Proof of Theorem 2oex

Step	Hyp	Ref	Expression
1		df2o3 8473	. 2 ⊢ 2o = {∅, 1o}
2		prex 5432	. 2 ⊢ {∅, 1o} ∈ V
3	1, 2	eqeltri 2829	1 ⊢ 2o ∈ V

Syntax hints:   ∈ wcel 2106  Vcvv 3474  ∅c0 4322  {cpr 4630  1_oc1o 8458  2_oc2o 8459

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-dif 3951  df-un 3953  df-nul 4323  df-sn 4629  df-pr 4631  df-suc 6370  df-1o 8465  df-2o 8466

This theorem is referenced by:  2on  8479  snnen2o  9236  1sdom2  9239  setc2obas  18043  setc2ohom  18044  nogt01o  27196  nosupbday  27205  noetainflem1  27237  noetainflem2  27238  noetainflem4  27240  fmlaomn0  34376  goaln0  34379  goalrlem  34382  goalr  34383  fmlasucdisj  34385  satffunlem1lem1  34388  satffunlem2lem1  34390  ex-sategoelel12  34413  oenord1ex  42055  onno  42174  clsk1indlem1  42786  clsk1independent  42787  setc2othin  47666