Theorem bj-1nel0 37451

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 8460	. . . 4 ⊢ 1o ≠ ∅
2	1	neii 2962	. . 3 ⊢ ¬ 1o = ∅
3		elsni 4602	. . 3 ⊢ (1o ∈ {∅} → 1o = ∅)
4	2, 3	mto 200	. 2 ⊢ ¬ 1o ∈ {∅}
5	4	nelir 3067	1 ⊢ 1o ∉ {∅}

Syntax hints:   = wceq 1563   ∈ wcel 2145   ∉ wnel 3064  ∅c0 4288  {csn 4585  1_oc1o 8434

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-nul 5261

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-nel 3065  df-v 3459  df-dif 3910  df-un 3912  df-nul 4289  df-sn 4586  df-suc 6356  df-1o 8441

This theorem is referenced by: (None)