Theorem bj-0nel1 37008

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 8412	. . . 4 ⊢ 1o ≠ ∅
2	1	nesymi 2987	. . 3 ⊢ ¬ ∅ = 1o
3		0ex 5249	. . . 4 ⊢ ∅ ∈ V
4	3	elsn 4592	. . 3 ⊢ (∅ ∈ {1o} ↔ ∅ = 1o)
5	2, 4	mtbir 323	. 2 ⊢ ¬ ∅ ∈ {1o}
6	5	nelir 3037	1 ⊢ ∅ ∉ {1o}

Syntax hints:   = wceq 1541   ∈ wcel 2113   ∉ wnel 3034  ∅c0 4284  {csn 4577  1_oc1o 8387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-nul 5248

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2931  df-nel 3035  df-v 3440  df-dif 3902  df-un 3904  df-nul 4285  df-sn 4578  df-suc 6320  df-1o 8394

This theorem is referenced by: (None)