Theorem bj-1nel0 36138

Description: 1_o does not belong to {∅}. (Contributed by BJ, 6-Apr-2019.)

Assertion

Ref	Expression
bj-1nel0	⊢ 1o ∉ {∅}

Proof of Theorem bj-1nel0

Step	Hyp	Ref	Expression
1		1n0 8490	. . . 4 ⊢ 1o ≠ ∅
2	1	neii 2940	. . 3 ⊢ ¬ 1o = ∅
3		elsni 4644	. . 3 ⊢ (1o ∈ {∅} → 1o = ∅)
4	2, 3	mto 196	. 2 ⊢ ¬ 1o ∈ {∅}
5	4	nelir 3047	1 ⊢ 1o ∉ {∅}

Syntax hints:   = wceq 1539   ∈ wcel 2104   ∉ wnel 3044  ∅c0 4321  {csn 4627  1_oc1o 8461

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-nul 5305

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-nel 3045  df-v 3474  df-dif 3950  df-un 3952  df-nul 4322  df-sn 4628  df-suc 6369  df-1o 8468

This theorem is referenced by: (None)