Theorem bj-0nel1 36976

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 8505	. . . 4 ⊢ 1o ≠ ∅
2	1	nesymi 2990	. . 3 ⊢ ¬ ∅ = 1o
3		0ex 5282	. . . 4 ⊢ ∅ ∈ V
4	3	elsn 4621	. . 3 ⊢ (∅ ∈ {1o} ↔ ∅ = 1o)
5	2, 4	mtbir 323	. 2 ⊢ ¬ ∅ ∈ {1o}
6	5	nelir 3040	1 ⊢ ∅ ∉ {1o}

Syntax hints:   = wceq 1540   ∈ wcel 2109   ∉ wnel 3037  ∅c0 4313  {csn 4606  1_oc1o 8478

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 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-nul 5281

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 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ne 2934  df-nel 3038  df-v 3466  df-dif 3934  df-un 3936  df-nul 4314  df-sn 4607  df-suc 6363  df-1o 8485

This theorem is referenced by: (None)