Theorem bj-1nel0 36942

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 8452	. . . 4 ⊢ 1o ≠ ∅
2	1	neii 2927	. . 3 ⊢ ¬ 1o = ∅
3		elsni 4606	. . 3 ⊢ (1o ∈ {∅} → 1o = ∅)
4	2, 3	mto 197	. 2 ⊢ ¬ 1o ∈ {∅}
5	4	nelir 3032	1 ⊢ 1o ∉ {∅}

Syntax hints:   = wceq 1540   ∈ wcel 2109   ∉ wnel 3029  ∅c0 4296  {csn 4589  1_oc1o 8427

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 2701  ax-nul 5261

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 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-nel 3030  df-v 3449  df-dif 3917  df-un 3919  df-nul 4297  df-sn 4590  df-suc 6338  df-1o 8434

This theorem is referenced by: (None)