Theorem bj-0nel1 32133

Description: The empty set does not belong to {1_𝑜}. (Contributed by BJ, 6-Apr-2019.)

Assertion

Ref	Expression
bj-0nel1	⊢ ∅ ∉ {1𝑜}

Proof of Theorem bj-0nel1

Step	Hyp	Ref	Expression
1		1n0 7462	. . . 4 ⊢ 1𝑜 ≠ ∅
2	1	nesymi 2839	. . 3 ⊢ ¬ ∅ = 1𝑜
3		0ex 4718	. . . 4 ⊢ ∅ ∈ V
4	3	elsn 4140	. . 3 ⊢ (∅ ∈ {1𝑜} ↔ ∅ = 1𝑜)
5	2, 4	mtbir 312	. 2 ⊢ ¬ ∅ ∈ {1𝑜}
6	5	nelir 2886	1 ⊢ ∅ ∉ {1𝑜}

Syntax hints:   = wceq 1475   ∈ wcel 1977   ∉ wnel 2781  ∅c0 3874  {csn 4125  1_𝑜c1o 7440

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-v 3175  df-dif 3543  df-un 3545  df-nul 3875  df-sn 4126  df-suc 5646  df-1o 7447

This theorem is referenced by: (None)