Theorem bj-0eltag 34293

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 5211	. . . . 5 ⊢ ∅ ∈ V
2	1	snid 4601	. . . 4 ⊢ ∅ ∈ {∅}
3	2	olci 862	. . 3 ⊢ (∅ ∈ sngl 𝐴 ∨ ∅ ∈ {∅})
4		elun 4125	. . 3 ⊢ (∅ ∈ (sngl 𝐴 ∪ {∅}) ↔ (∅ ∈ sngl 𝐴 ∨ ∅ ∈ {∅}))
5	3, 4	mpbir 233	. 2 ⊢ ∅ ∈ (sngl 𝐴 ∪ {∅})
6		df-bj-tag 34290	. 2 ⊢ tag 𝐴 = (sngl 𝐴 ∪ {∅})
7	5, 6	eleqtrri 2912	1 ⊢ ∅ ∈ tag 𝐴

Syntax hints:   ∨ wo 843   ∈ wcel 2114   ∪ cun 3934  ∅c0 4291  {csn 4567  sngl bj-csngl 34280  tag bj-ctag 34289

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-nul 5210

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3496  df-dif 3939  df-un 3941  df-nul 4292  df-sn 4568  df-bj-tag 34290

This theorem is referenced by:  bj-tagn0  34294