Theorem bj-0nel1 34265

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 8118	. . . 4 ⊢ 1o ≠ ∅
2	1	nesymi 3073	. . 3 ⊢ ¬ ∅ = 1o
3		0ex 5210	. . . 4 ⊢ ∅ ∈ V
4	3	elsn 4581	. . 3 ⊢ (∅ ∈ {1o} ↔ ∅ = 1o)
5	2, 4	mtbir 325	. 2 ⊢ ¬ ∅ ∈ {1o}
6	5	nelir 3126	1 ⊢ ∅ ∉ {1o}

Syntax hints:   = wceq 1533   ∈ wcel 2110   ∉ wnel 3123  ∅c0 4290  {csn 4566  1_oc1o 8094

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-nul 5209

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-v 3496  df-dif 3938  df-un 3940  df-nul 4291  df-sn 4567  df-suc 6196  df-1o 8101

This theorem is referenced by: (None)