Theorem bj-0nel1 37958

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 8498	. . . 4 ⊢ 1o ≠ ∅
2	1	nesymi 3012	. . 3 ⊢ ¬ ∅ = 1o
3		0ex 5260	. . . 4 ⊢ ∅ ∈ V
4	3	elsn 4595	. . 3 ⊢ (∅ ∈ {1o} ↔ ∅ = 1o)
5	2, 4	mtbir 326	. 2 ⊢ ¬ ∅ ∈ {1o}
6	5	nelir 3064	1 ⊢ ∅ ∉ {1o}

This proof depends on syntax axioms:   = wceq 1569   ∈ wcel 2144   ∉ wnel 3061  ∅c0 4278  {csn 4584  1_oc1o 8472

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-nul 5259

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-nel 3062  df-v 3452  df-dif 4047  df-un 4151  df-nul 4279  df-sn 4585  df-suc 6368  df-1o 8479

This theorem is used by: (None)