Theorem bj-0nel1 34766

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 8150	. . . 4 ⊢ 1o ≠ ∅
2	1	nesymi 2991	. . 3 ⊢ ¬ ∅ = 1o
3		0ex 5175	. . . 4 ⊢ ∅ ∈ V
4	3	elsn 4531	. . 3 ⊢ (∅ ∈ {1o} ↔ ∅ = 1o)
5	2, 4	mtbir 326	. 2 ⊢ ¬ ∅ ∈ {1o}
6	5	nelir 3041	1 ⊢ ∅ ∉ {1o}

Syntax hints:   = wceq 1542   ∈ wcel 2114   ∉ wnel 3038  ∅c0 4211  {csn 4516  1_oc1o 8124

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710  ax-nul 5174

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ne 2935  df-nel 3039  df-v 3400  df-dif 3846  df-un 3848  df-nul 4212  df-sn 4517  df-suc 6178  df-1o 8131

This theorem is referenced by: (None)