Theorem bj-1nel0 34268

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 8121	. . . 4 ⊢ 1o ≠ ∅
2	1	neii 3020	. . 3 ⊢ ¬ 1o = ∅
3		elsni 4586	. . 3 ⊢ (1o ∈ {∅} → 1o = ∅)
4	2, 3	mto 199	. 2 ⊢ ¬ 1o ∈ {∅}
5	4	nelir 3128	1 ⊢ 1o ∉ {∅}

Syntax hints:   = wceq 1537   ∈ wcel 2114   ∉ wnel 3125  ∅c0 4293  {csn 4569  1_oc1o 8097

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-nul 5212

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-v 3498  df-dif 3941  df-un 3943  df-nul 4294  df-sn 4570  df-suc 6199  df-1o 8104

This theorem is referenced by: (None)