Theorem 2oex 8481

Description: 2_o is a set. (Contributed by BJ, 6-Apr-2019.) Remove dependency on ax-10 2178, ax-11 2194, ax-12 2213, ax-un 7749. (Proof shortened by Zhi Wang, 19-Sep-2024.)

Assertion

Ref	Expression
2oex	⊢ 2o ∈ V

Proof of Theorem 2oex

Step	Hyp	Ref	Expression
1		df2o3 8477	. 2 ⊢ 2o = {∅, 1o}
2		prex 5396	. 2 ⊢ {∅, 1o} ∈ V
3	1, 2	eqeltri 2857	1 ⊢ 2o ∈ V

This proof depends on syntax axioms:   ∈ wcel 2145  Vcvv 3451  ∅c0 4279  {cpr 4586  1_oc1o 8462  2_oc2o 8463

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3453  df-dif 3902  df-un 3904  df-nul 4280  df-sn 4585  df-pr 4587  df-suc 6367  df-1o 8469  df-2o 8470

This theorem is used by:  2on  8483  snnen2o  9229  1sdom2  9232  setc2obas  18262  setc2ohom  18263  degenmgmopdm  19127  degenmgm  19130  degenmgm2opdm  19131  degenmgm2nfun  19132  degenmgm2  19133  nogt01o  28046  nosupbday  28055  noetainflem1  28087  noetainflem2  28088  noetainflem4  28090  fmlaomn0  36134  goaln0  36137  goalrlem  36140  goalr  36141  fmlasucdisj  36143  satffunlem1lem1  36146  satffunlem2lem1  36148  ex-sategoelel12  36171  oenord1ex  44301  onnoxp  44418  clsk1indlem1  45030  clsk1independent  45031  nelsubc3  50148  setc2othin  50543  setc1onsubc  50679