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Theorem bj-eltag 33833
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
StepHypRef Expression
1 df-bj-tag 33831 . . 3  |- tag  B  =  (sngl  B  u.  { (/)
} )
21eleq2i 2545 . 2  |-  ( A  e. tag  B  <->  A  e.  (sngl  B  u.  { (/) } ) )
3 elun 3645 . 2  |-  ( A  e.  (sngl  B  u.  {
(/) } )  <->  ( A  e. sngl  B  \/  A  e. 
{ (/) } ) )
4 bj-elsngl 33824 . . 3  |-  ( A  e. sngl  B  <->  E. x  e.  B  A  =  { x } )
5 0ex 4577 . . . 4  |-  (/)  e.  _V
65elsnc2 4058 . . 3  |-  ( A  e.  { (/) }  <->  A  =  (/) )
74, 6orbi12i 521 . 2  |-  ( ( A  e. sngl  B  \/  A  e.  { (/) } )  <-> 
( E. x  e.  B  A  =  {
x }  \/  A  =  (/) ) )
82, 3, 73bitri 271 1  |-  ( A  e. tag  B  <->  ( E. x  e.  B  A  =  { x }  \/  A  =  (/) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    \/ wo 368    = wceq 1379    e. wcel 1767   E.wrex 2815    u. cun 3474   (/)c0 3785   {csn 4027  sngl bj-csngl 33821  tag bj-ctag 33830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-v 3115  df-dif 3479  df-un 3481  df-nul 3786  df-sn 4028  df-pr 4030  df-bj-sngl 33822  df-bj-tag 33831
This theorem is referenced by: (None)
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