Theorem 2oex 8491

Description: 2_o is a set. (Contributed by BJ, 6-Apr-2019.) Remove dependency on ax-10 2141, ax-11 2157, ax-12 2177, 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 8488	. 2 ⊢ 2o = {∅, 1o}
2		prex 5407	. 2 ⊢ {∅, 1o} ∈ V
3	1, 2	eqeltri 2830	1 ⊢ 2o ∈ V

Syntax hints:   ∈ wcel 2108  Vcvv 3459  ∅c0 4308  {cpr 4603  1_oc1o 8473  2_oc2o 8474

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 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

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 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-v 3461  df-dif 3929  df-un 3931  df-nul 4309  df-sn 4602  df-pr 4604  df-suc 6358  df-1o 8480  df-2o 8481

This theorem is referenced by:  2on  8494  snnen2o  9245  1sdom2  9248  setc2obas  18107  setc2ohom  18108  nogt01o  27660  nosupbday  27669  noetainflem1  27701  noetainflem2  27702  noetainflem4  27704  fmlaomn0  35412  goaln0  35415  goalrlem  35418  goalr  35419  fmlasucdisj  35421  satffunlem1lem1  35424  satffunlem2lem1  35426  ex-sategoelel12  35449  oenord1ex  43339  onno  43457  clsk1indlem1  44069  clsk1independent  44070  nelsubc3  49038  setc2othin  49352  setc1onsubc  49479