Theorem bj-0nel1 37705

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 8478	. . . 4 ⊢ 1o ≠ ∅
2	1	nesymi 3014	. . 3 ⊢ ¬ ∅ = 1o
3		0ex 5268	. . . 4 ⊢ ∅ ∈ V
4	3	elsn 4602	. . 3 ⊢ (∅ ∈ {1o} ↔ ∅ = 1o)
5	2, 4	mtbir 326	. 2 ⊢ ¬ ∅ ∈ {1o}
6	5	nelir 3066	1 ⊢ ∅ ∉ {1o}

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2145   ∉ wnel 3063  ∅c0 4282  {csn 4587  1_oc1o 8452

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-nul 5267

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 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-nel 3064  df-v 3455  df-dif 3905  df-un 3907  df-nul 4283  df-sn 4588  df-suc 6367  df-1o 8459

This theorem is used by: (None)