Theorem bnj105 31130

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Assertion

Ref	Expression
bnj105	⊢ 1𝑜 ∈ V

Proof of Theorem bnj105

Step	Hyp	Ref	Expression
1		df1o2 7726	. 2 ⊢ 1𝑜 = {∅}
2		p0ex 4984	. 2 ⊢ {∅} ∈ V
3	1, 2	eqeltri 2846	1 ⊢ 1𝑜 ∈ V

Syntax hints:   ∈ wcel 2145  Vcvv 3351  ∅c0 4063  {csn 4316  1_𝑜c1o 7706

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-pw 4299  df-sn 4317  df-suc 5872  df-1o 7713

This theorem is referenced by:  bnj106  31276  bnj118  31277  bnj121  31278  bnj125  31280  bnj130  31282  bnj153  31288