Theorem 2oex 8516

Description: 2_o is a set. (Contributed by BJ, 6-Apr-2019.) Remove dependency on ax-10 2139, ax-11 2155, ax-12 2175, ax-un 7754. (Proof shortened by Zhi Wang, 19-Sep-2024.)

Assertion

Ref	Expression
2oex	⊢ 2o ∈ V

Proof of Theorem 2oex

Step	Hyp	Ref	Expression
1		df2o3 8513	. 2 ⊢ 2o = {∅, 1o}
2		prex 5443	. 2 ⊢ {∅, 1o} ∈ V
3	1, 2	eqeltri 2835	1 ⊢ 2o ∈ V

Syntax hints:   ∈ wcel 2106  Vcvv 3478  ∅c0 4339  {cpr 4633  1_oc1o 8498  2_oc2o 8499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-dif 3966  df-un 3968  df-nul 4340  df-sn 4632  df-pr 4634  df-suc 6392  df-1o 8505  df-2o 8506

This theorem is referenced by:  2on  8519  snnen2o  9271  1sdom2  9274  setc2obas  18148  setc2ohom  18149  nogt01o  27756  nosupbday  27765  noetainflem1  27797  noetainflem2  27798  noetainflem4  27800  fmlaomn0  35375  goaln0  35378  goalrlem  35381  goalr  35382  fmlasucdisj  35384  satffunlem1lem1  35387  satffunlem2lem1  35389  ex-sategoelel12  35412  oenord1ex  43305  onno  43423  clsk1indlem1  44035  clsk1independent  44036  setc2othin  48857