Theorem bj-0eltag 37642

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 5269	. . . . 5 ⊢ ∅ ∈ V
2	1	snid 4627	. . . 4 ⊢ ∅ ∈ {∅}
3	2	olci 879	. . 3 ⊢ (∅ ∈ sngl 𝐴 ∨ ∅ ∈ {∅})
4		elun 4106	. . 3 ⊢ (∅ ∈ (sngl 𝐴 ∪ {∅}) ↔ (∅ ∈ sngl 𝐴 ∨ ∅ ∈ {∅}))
5	3, 4	mpbir 234	. 2 ⊢ ∅ ∈ (sngl 𝐴 ∪ {∅})
6		df-bj-tag 37639	. 2 ⊢ tag 𝐴 = (sngl 𝐴 ∪ {∅})
7	5, 6	eleqtrri 2861	1 ⊢ ∅ ∈ tag 𝐴

This proof depends on syntax axioms:   ∨ wo 860   ∈ wcel 2142   ∪ cun 3902  ∅c0 4285  {csn 4588  sngl bj-csngl 37629  tag bj-ctag 37638

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-nul 5268

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3456  df-dif 3907  df-un 3909  df-nul 4286  df-sn 4589  df-bj-tag 37639

This theorem is used by:  bj-tagn0  37643