Theorem bj-0nel1 38044

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 8523	. . . 4 ⊢ 1o ≠ ∅
2	1	nesymi 3016	. . 3 ⊢ ¬ ∅ = 1o
3		0ex 5264	. . . 4 ⊢ ∅ ∈ V
4	3	elsn 4599	. . 3 ⊢ (∅ ∈ {1o} ↔ ∅ = 1o)
5	2, 4	mtbir 326	. 2 ⊢ ¬ ∅ ∈ {1o}
6	5	nelir 3068	1 ⊢ ∅ ∉ {1o}

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2147   ∉ wnel 3065  ∅c0 4281  {csn 4588  1_oc1o 8497

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-nul 5263

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-nel 3066  df-v 3456  df-dif 4050  df-un 4154  df-nul 4282  df-sn 4589  df-suc 6386  df-1o 8504

This theorem is used by: (None)