Theorem bj-0nel1 36936

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 8525	. . . 4 ⊢ 1o ≠ ∅
2	1	nesymi 2996	. . 3 ⊢ ¬ ∅ = 1o
3		0ex 5313	. . . 4 ⊢ ∅ ∈ V
4	3	elsn 4646	. . 3 ⊢ (∅ ∈ {1o} ↔ ∅ = 1o)
5	2, 4	mtbir 323	. 2 ⊢ ¬ ∅ ∈ {1o}
6	5	nelir 3047	1 ⊢ ∅ ∉ {1o}

Syntax hints:   = wceq 1537   ∈ wcel 2106   ∉ wnel 3044  ∅c0 4339  {csn 4631  1_oc1o 8498

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-nul 5312

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-nel 3045  df-v 3480  df-dif 3966  df-un 3968  df-nul 4340  df-sn 4632  df-suc 6392  df-1o 8505

This theorem is referenced by: (None)