Theorem bj-0nel1 36138

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 8492	. . . 4 ⊢ 1o ≠ ∅
2	1	nesymi 2997	. . 3 ⊢ ¬ ∅ = 1o
3		0ex 5307	. . . 4 ⊢ ∅ ∈ V
4	3	elsn 4643	. . 3 ⊢ (∅ ∈ {1o} ↔ ∅ = 1o)
5	2, 4	mtbir 323	. 2 ⊢ ¬ ∅ ∈ {1o}
6	5	nelir 3048	1 ⊢ ∅ ∉ {1o}

Syntax hints:   = wceq 1540   ∈ wcel 2105   ∉ wnel 3045  ∅c0 4322  {csn 4628  1_oc1o 8463

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-nul 5306

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-nel 3046  df-v 3475  df-dif 3951  df-un 3953  df-nul 4323  df-sn 4629  df-suc 6370  df-1o 8470

This theorem is referenced by: (None)