Theorem bj-0nel1 37555

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 8471	. . . 4 ⊢ 1o ≠ ∅
2	1	nesymi 3013	. . 3 ⊢ ¬ ∅ = 1o
3		0ex 5269	. . . 4 ⊢ ∅ ∈ V
4	3	elsn 4603	. . 3 ⊢ (∅ ∈ {1o} ↔ ∅ = 1o)
5	2, 4	mtbir 326	. 2 ⊢ ¬ ∅ ∈ {1o}
6	5	nelir 3065	1 ⊢ ∅ ∉ {1o}

Syntax hints:   = wceq 1568   ∈ wcel 2141   ∉ wnel 3062  ∅c0 4285  {csn 4588  1_oc1o 8445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733  ax-nul 5268

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-nel 3063  df-v 3455  df-dif 3907  df-un 3909  df-nul 4286  df-sn 4589  df-suc 6366  df-1o 8452

This theorem is referenced by: (None)