Theorem 2oex 8448

Description: 2_o is a set. (Contributed by BJ, 6-Apr-2019.) Remove dependency on ax-10 2142, ax-11 2158, ax-12 2178, ax-un 7714. (Proof shortened by Zhi Wang, 19-Sep-2024.)

Assertion

Ref	Expression
2oex	⊢ 2o ∈ V

Proof of Theorem 2oex

Step	Hyp	Ref	Expression
1		df2o3 8445	. 2 ⊢ 2o = {∅, 1o}
2		prex 5395	. 2 ⊢ {∅, 1o} ∈ V
3	1, 2	eqeltri 2825	1 ⊢ 2o ∈ V

Syntax hints:   ∈ wcel 2109  Vcvv 3450  ∅c0 4299  {cpr 4594  1_oc1o 8430  2_oc2o 8431

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 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3452  df-dif 3920  df-un 3922  df-nul 4300  df-sn 4593  df-pr 4595  df-suc 6341  df-1o 8437  df-2o 8438

This theorem is referenced by:  2on  8450  snnen2o  9191  1sdom2  9194  setc2obas  18063  setc2ohom  18064  nogt01o  27615  nosupbday  27624  noetainflem1  27656  noetainflem2  27657  noetainflem4  27659  fmlaomn0  35384  goaln0  35387  goalrlem  35390  goalr  35391  fmlasucdisj  35393  satffunlem1lem1  35396  satffunlem2lem1  35398  ex-sategoelel12  35421  oenord1ex  43311  onno  43429  clsk1indlem1  44041  clsk1independent  44042  nelsubc3  49064  setc2othin  49459  setc1onsubc  49595