Theorem 2oex 8449

Description: 2_o is a set. (Contributed by BJ, 6-Apr-2019.) Remove dependency on ax-10 2175, ax-11 2191, ax-12 2212, ax-un 7718. (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 2858	1 ⊢ 2o ∈ V

Syntax hints:   ∈ wcel 2142  Vcvv 3454  ∅c0 4285  {cpr 4584  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 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3456  df-dif 3907  df-un 3909  df-nul 4286  df-sn 4583  df-pr 4585  df-suc 6352  df-1o 8437  df-2o 8438

This theorem is referenced by:  2on  8451  snnen2o  9189  1sdom2  9192  setc2obas  18127  setc2ohom  18128  nogt01o  27757  nosupbday  27766  noetainflem1  27798  noetainflem2  27799  noetainflem4  27801  fmlaomn0  35737  goaln0  35740  goalrlem  35743  goalr  35744  fmlasucdisj  35746  satffunlem1lem1  35749  satffunlem2lem1  35751  ex-sategoelel12  35774  oenord1ex  43889  onnoxp  44006  clsk1indlem1  44618  clsk1independent  44619  nelsubc3  49689  setc2othin  50084  setc1onsubc  50220