Theorem bj-0eltag 36944

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 5325	. . . . 5 ⊢ ∅ ∈ V
2	1	snid 4684	. . . 4 ⊢ ∅ ∈ {∅}
3	2	olci 865	. . 3 ⊢ (∅ ∈ sngl 𝐴 ∨ ∅ ∈ {∅})
4		elun 4176	. . 3 ⊢ (∅ ∈ (sngl 𝐴 ∪ {∅}) ↔ (∅ ∈ sngl 𝐴 ∨ ∅ ∈ {∅}))
5	3, 4	mpbir 231	. 2 ⊢ ∅ ∈ (sngl 𝐴 ∪ {∅})
6		df-bj-tag 36941	. 2 ⊢ tag 𝐴 = (sngl 𝐴 ∪ {∅})
7	5, 6	eleqtrri 2843	1 ⊢ ∅ ∈ tag 𝐴

Syntax hints:   ∨ wo 846   ∈ wcel 2108   ∪ cun 3974  ∅c0 4352  {csn 4648  sngl bj-csngl 36931  tag bj-ctag 36940

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-nul 5324

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-v 3490  df-dif 3979  df-un 3981  df-nul 4353  df-sn 4649  df-bj-tag 36941

This theorem is referenced by:  bj-tagn0  36945