Theorem bj-0eltag 37891

Description: The empty set belongs to the tagging of a class. (Contributed by BJ, 6-Apr-2019.)

Assertion

Ref	Expression
bj-0eltag	⊢ ∅ ∈ tag 𝐴

Proof of Theorem bj-0eltag

Step	Hyp	Ref	Expression
1		0ex 5260	. . . . 5 ⊢ ∅ ∈ V
2	1	snid 4622	. . . 4 ⊢ ∅ ∈ {∅}
3	2	olci 880	. . 3 ⊢ (∅ ∈ sngl 𝐴 ∨ ∅ ∈ {∅})
4		elun 4152	. . 3 ⊢ (∅ ∈ (sngl 𝐴 ∪ {∅}) ↔ (∅ ∈ sngl 𝐴 ∨ ∅ ∈ {∅}))
5	3, 4	mpbir 234	. 2 ⊢ ∅ ∈ (sngl 𝐴 ∪ {∅})
6		df-bj-tag 37888	. 2 ⊢ tag 𝐴 = (sngl 𝐴 ∪ {∅})
7	5, 6	eleqtrri 2859	1 ⊢ ∅ ∈ tag 𝐴

This proof depends on syntax axioms:   ∨ wo 861   ∈ wcel 2144   ∪ cun 4149  ∅c0 4278  {csn 4583  sngl bj-csngl 37878  tag bj-ctag 37887

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-nul 5259

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-dif 4047  df-un 4151  df-nul 4279  df-sn 4584  df-bj-tag 37888

This theorem is used by:  bj-tagn0  37892