Theorem bj-1nel0 36998

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 8403	. . . 4 ⊢ 1o ≠ ∅
2	1	neii 2930	. . 3 ⊢ ¬ 1o = ∅
3		elsni 4590	. . 3 ⊢ (1o ∈ {∅} → 1o = ∅)
4	2, 3	mto 197	. 2 ⊢ ¬ 1o ∈ {∅}
5	4	nelir 3035	1 ⊢ 1o ∉ {∅}

Syntax hints:   = wceq 1541   ∈ wcel 2111   ∉ wnel 3032  ∅c0 4280  {csn 4573  1_oc1o 8378

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-nul 5242

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-nel 3033  df-v 3438  df-dif 3900  df-un 3902  df-nul 4281  df-sn 4574  df-suc 6312  df-1o 8385

This theorem is referenced by: (None)