Theorem bj-1nel0 36977

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 8505	. . . 4 ⊢ 1o ≠ ∅
2	1	neii 2935	. . 3 ⊢ ¬ 1o = ∅
3		elsni 4623	. . 3 ⊢ (1o ∈ {∅} → 1o = ∅)
4	2, 3	mto 197	. 2 ⊢ ¬ 1o ∈ {∅}
5	4	nelir 3040	1 ⊢ 1o ∉ {∅}

Syntax hints:   = wceq 1540   ∈ wcel 2109   ∉ wnel 3037  ∅c0 4313  {csn 4606  1_oc1o 8478

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-nul 5281

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ne 2934  df-nel 3038  df-v 3466  df-dif 3934  df-un 3936  df-nul 4314  df-sn 4607  df-suc 6363  df-1o 8485

This theorem is referenced by: (None)