Theorem bj-0eltag 32966

Description: The empty set belongs to the tagging of a class. (Contributed by BJ, 6-Apr-2019.)

Assertion

Ref	Expression
bj-0eltag	|- (/) e. tag A

Proof of Theorem bj-0eltag

Step	Hyp	Ref	Expression
1		0ex 4790	. . . . 5 |- (/) e. _V
2	1	snid 4208	. . . 4 |- (/) e. { (/) }
3	2	olci 406	. . 3 |- ( (/) e. sngl A \/ (/) e. { (/) } )
4		elun 3753	. . 3 |- ( (/) e. (sngl A u. { (/) } ) <-> ( (/) e. sngl A \/ (/) e. { (/) } ) )
5	3, 4	mpbir 221	. 2 |- (/) e. (sngl A u. { (/) } )
6		df-bj-tag 32963	. 2 |- tag A = (sngl A u. { (/) } )
7	5, 6	eleqtrri 2700	1 |- (/) e. tag A

Syntax hints:   \/ wo 383   e. wcel 1990   u. cun 3572   (/)c0 3915   {csn 4177  sngl bj-csngl 32953  tag bj-ctag 32962

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577  df-un 3579  df-nul 3916  df-sn 4178  df-bj-tag 32963

This theorem is referenced by:  bj-tagn0  32967