Theorem bj-0eltag 36979

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 5307	. . . . 5 ⊢ ∅ ∈ V
2	1	snid 4662	. . . 4 ⊢ ∅ ∈ {∅}
3	2	olci 867	. . 3 ⊢ (∅ ∈ sngl 𝐴 ∨ ∅ ∈ {∅})
4		elun 4153	. . 3 ⊢ (∅ ∈ (sngl 𝐴 ∪ {∅}) ↔ (∅ ∈ sngl 𝐴 ∨ ∅ ∈ {∅}))
5	3, 4	mpbir 231	. 2 ⊢ ∅ ∈ (sngl 𝐴 ∪ {∅})
6		df-bj-tag 36976	. 2 ⊢ tag 𝐴 = (sngl 𝐴 ∪ {∅})
7	5, 6	eleqtrri 2840	1 ⊢ ∅ ∈ tag 𝐴

Syntax hints:   ∨ wo 848   ∈ wcel 2108   ∪ cun 3949  ∅c0 4333  {csn 4626  sngl bj-csngl 36966  tag bj-ctag 36975

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-dif 3954  df-un 3956  df-nul 4334  df-sn 4627  df-bj-tag 36976

This theorem is referenced by:  bj-tagn0  36980