Theorem bj-0eltag 37346

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 5232	. . . . 5 ⊢ ∅ ∈ V
2	1	snid 4597	. . . 4 ⊢ ∅ ∈ {∅}
3	2	olci 873	. . 3 ⊢ (∅ ∈ sngl 𝐴 ∨ ∅ ∈ {∅})
4		elun 4086	. . 3 ⊢ (∅ ∈ (sngl 𝐴 ∪ {∅}) ↔ (∅ ∈ sngl 𝐴 ∨ ∅ ∈ {∅}))
5	3, 4	mpbir 233	. 2 ⊢ ∅ ∈ (sngl 𝐴 ∪ {∅})
6		df-bj-tag 37343	. 2 ⊢ tag 𝐴 = (sngl 𝐴 ∪ {∅})
7	5, 6	eleqtrri 2840	1 ⊢ ∅ ∈ tag 𝐴

Syntax hints:   ∨ wo 854   ∈ wcel 2121   ∪ cun 3883  ∅c0 4264  {csn 4558  sngl bj-csngl 37333  tag bj-ctag 37342

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-nul 5231

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-v 3435  df-dif 3888  df-un 3890  df-nul 4265  df-sn 4559  df-bj-tag 37343

This theorem is referenced by:  bj-tagn0  37347