Theorem bj-0nel1 37622

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 8481	. . . 4 ⊢ 1o ≠ ∅
2	1	nesymi 3018	. . 3 ⊢ ¬ ∅ = 1o
3		0ex 5275	. . . 4 ⊢ ∅ ∈ V
4	3	elsn 4609	. . 3 ⊢ (∅ ∈ {1o} ↔ ∅ = 1o)
5	2, 4	mtbir 326	. 2 ⊢ ¬ ∅ ∈ {1o}
6	5	nelir 3070	1 ⊢ ∅ ∉ {1o}

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2146   ∉ wnel 3067  ∅c0 4289  {csn 4594  1_oc1o 8455

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 2148  ax-9 2156  ax-ext 2738  ax-nul 5274

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 2745  df-cleq 2758  df-clel 2841  df-ne 2962  df-nel 3068  df-v 3460  df-dif 3911  df-un 3913  df-nul 4290  df-sn 4595  df-suc 6373  df-1o 8462

This theorem is used by: (None)