Theorem bj-0nel1 35070

Description: The empty set does not belong to {1_o}. (Contributed by BJ, 6-Apr-2019.)

Assertion

Ref	Expression
bj-0nel1	⊢ ∅ ∉ {1o}

Proof of Theorem bj-0nel1

Step	Hyp	Ref	Expression
1		1n0 8286	. . . 4 ⊢ 1o ≠ ∅
2	1	nesymi 3000	. . 3 ⊢ ¬ ∅ = 1o
3		0ex 5226	. . . 4 ⊢ ∅ ∈ V
4	3	elsn 4573	. . 3 ⊢ (∅ ∈ {1o} ↔ ∅ = 1o)
5	2, 4	mtbir 322	. 2 ⊢ ¬ ∅ ∈ {1o}
6	5	nelir 3051	1 ⊢ ∅ ∉ {1o}

Syntax hints:   = wceq 1539   ∈ wcel 2108   ∉ wnel 3048  ∅c0 4253  {csn 4558  1_oc1o 8260

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-nul 5225

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-nel 3049  df-v 3424  df-dif 3886  df-un 3888  df-nul 4254  df-sn 4559  df-suc 6257  df-1o 8267

This theorem is referenced by: (None)