Theorem bj-0eltag 37562

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 5275	. . . . 5 ⊢ ∅ ∈ V
2	1	snid 4633	. . . 4 ⊢ ∅ ∈ {∅}
3	2	olci 879	. . 3 ⊢ (∅ ∈ sngl 𝐴 ∨ ∅ ∈ {∅})
4		elun 4115	. . 3 ⊢ (∅ ∈ (sngl 𝐴 ∪ {∅}) ↔ (∅ ∈ sngl 𝐴 ∨ ∅ ∈ {∅}))
5	3, 4	mpbir 234	. 2 ⊢ ∅ ∈ (sngl 𝐴 ∪ {∅})
6		df-bj-tag 37559	. 2 ⊢ tag 𝐴 = (sngl 𝐴 ∪ {∅})
7	5, 6	eleqtrri 2869	1 ⊢ ∅ ∈ tag 𝐴

Syntax hints:   ∨ wo 860   ∈ wcel 2150   ∪ cun 3911  ∅c0 4294  {csn 4594  sngl bj-csngl 37549  tag bj-ctag 37558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-ext 2742  ax-nul 5274

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-v 3464  df-dif 3916  df-un 3918  df-nul 4295  df-sn 4595  df-bj-tag 37559

This theorem is referenced by:  bj-tagn0  37563