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Theorem bj-eltag 31320
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 31318 . . 3  |- tag  B  =  (sngl  B  u.  { (/)
} )
21eleq2i 2507 . 2  |-  ( A  e. tag  B  <->  A  e.  (sngl  B  u.  { (/) } ) )
3 elun 3612 . 2  |-  ( A  e.  (sngl  B  u.  {
(/) } )  <->  ( A  e. sngl  B  \/  A  e. 
{ (/) } ) )
4 bj-elsngl 31311 . . 3  |-  ( A  e. sngl  B  <->  E. x  e.  B  A  =  { x } )
5 0ex 4557 . . . 4  |-  (/)  e.  _V
65elsnc2 4033 . . 3  |-  ( A  e.  { (/) }  <->  A  =  (/) )
74, 6orbi12i 523 . 2  |-  ( ( A  e. sngl  B  \/  A  e.  { (/) } )  <-> 
( E. x  e.  B  A  =  {
x }  \/  A  =  (/) ) )
82, 3, 73bitri 274 1  |-  ( A  e. tag  B  <->  ( E. x  e.  B  A  =  { x }  \/  A  =  (/) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    \/ wo 369    = wceq 1437    e. wcel 1870   E.wrex 2783    u. cun 3440   (/)c0 3767   {csn 4002  sngl bj-csngl 31308  tag bj-ctag 31317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-v 3089  df-dif 3445  df-un 3447  df-nul 3768  df-sn 4003  df-pr 4005  df-bj-sngl 31309  df-bj-tag 31318
This theorem is referenced by: (None)
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