Theorem bj-1nel0 37556

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 8471	. . . 4 ⊢ 1o ≠ ∅
2	1	neii 2958	. . 3 ⊢ ¬ 1o = ∅
3		elsni 4605	. . 3 ⊢ (1o ∈ {∅} → 1o = ∅)
4	2, 3	mto 200	. 2 ⊢ ¬ 1o ∈ {∅}
5	4	nelir 3065	1 ⊢ 1o ∉ {∅}

Syntax hints:   = wceq 1568   ∈ wcel 2141   ∉ wnel 3062  ∅c0 4285  {csn 4588  1_oc1o 8445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733  ax-nul 5268

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-nel 3063  df-v 3455  df-dif 3907  df-un 3909  df-nul 4286  df-sn 4589  df-suc 6366  df-1o 8452

This theorem is referenced by: (None)