Theorem 2oex 8483

Description: 2_o is a set. (Contributed by BJ, 6-Apr-2019.) Remove dependency on ax-10 2136, ax-11 2153, ax-12 2170, ax-un 7729. (Proof shortened by Zhi Wang, 19-Sep-2024.)

Assertion

Ref	Expression
2oex	⊢ 2o ∈ V

Proof of Theorem 2oex

Step	Hyp	Ref	Expression
1		df2o3 8480	. 2 ⊢ 2o = {∅, 1o}
2		prex 5432	. 2 ⊢ {∅, 1o} ∈ V
3	1, 2	eqeltri 2828	1 ⊢ 2o ∈ V

Syntax hints:   ∈ wcel 2105  Vcvv 3473  ∅c0 4322  {cpr 4630  1_oc1o 8465  2_oc2o 8466

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3475  df-dif 3951  df-un 3953  df-nul 4323  df-sn 4629  df-pr 4631  df-suc 6370  df-1o 8472  df-2o 8473

This theorem is referenced by:  2on  8486  snnen2o  9243  1sdom2  9246  setc2obas  18051  setc2ohom  18052  nogt01o  27450  nosupbday  27459  noetainflem1  27491  noetainflem2  27492  noetainflem4  27494  fmlaomn0  34694  goaln0  34697  goalrlem  34700  goalr  34701  fmlasucdisj  34703  satffunlem1lem1  34706  satffunlem2lem1  34708  ex-sategoelel12  34731  oenord1ex  42380  onno  42499  clsk1indlem1  43111  clsk1independent  43112  setc2othin  47776