Theorem bnj105 30044

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 7459	. 2 ⊢ 1𝑜 = {∅}
2		p0ex 4779	. 2 ⊢ {∅} ∈ V
3	1, 2	eqeltri 2684	1 ⊢ 1𝑜 ∈ V

Syntax hints:   ∈ wcel 1977  Vcvv 3173  ∅c0 3874  {csn 4125  1_𝑜c1o 7440

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-pw 4110  df-sn 4126  df-suc 5646  df-1o 7447

This theorem is referenced by:  bnj106  30192  bnj118  30193  bnj121  30194  bnj125  30196  bnj130  30198  bnj153  30204