Theorem bj-eltag 32965

Description: Characterization of the elements of the tagging of a class. (Contributed by BJ, 6-Oct-2018.)

Assertion

Ref	Expression
bj-eltag	|- ( A e. tag B <-> ( E. x e. B A = { x } \/ A = (/) ) )

Distinct variable groups:   x, A   x, B

Proof of Theorem bj-eltag

Step	Hyp	Ref	Expression
1		df-bj-tag 32963	. . 3 |- tag B = (sngl B u. { (/) } )
2	1	eleq2i 2693	. 2 |- ( A e. tag B <-> A e. (sngl B u. { (/) } ) )
3		elun 3753	. 2 |- ( A e. (sngl B u. { (/) } ) <-> ( A e. sngl B \/ A e. { (/) } ) )
4		bj-elsngl 32956	. . 3 |- ( A e. sngl B <-> E. x e. B A = { x } )
5		0ex 4790	. . . 4 |- (/) e. _V
6	5	elsn2 4211	. . 3 |- ( A e. { (/) } <-> A = (/) )
7	4, 6	orbi12i 543	. 2 |- ( ( A e. sngl B \/ A e. { (/) } ) <-> ( E. x e. B A = { x } \/ A = (/) ) )
8	2, 3, 7	3bitri 286	1 |- ( A e. tag B <-> ( E. x e. B A = { x } \/ A = (/) ) )

Syntax hints:   <-> wb 196   \/ wo 383   = wceq 1483   e. wcel 1990   E.wrex 2913   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-sep 4781  ax-nul 4789  ax-pr 4906

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-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-un 3579  df-nul 3916  df-sn 4178  df-pr 4180  df-bj-sngl 32954  df-bj-tag 32963

This theorem is referenced by: (None)