Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-eltag Structured version   Unicode version

Theorem bj-eltag 32803
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 32801 . . 3  |- tag  B  =  (sngl  B  u.  { (/)
} )
21eleq2i 2532 . 2  |-  ( A  e. tag  B  <->  A  e.  (sngl  B  u.  { (/) } ) )
3 elun 3606 . 2  |-  ( A  e.  (sngl  B  u.  {
(/) } )  <->  ( A  e. sngl  B  \/  A  e. 
{ (/) } ) )
4 bj-elsngl 32794 . . 3  |-  ( A  e. sngl  B  <->  E. x  e.  B  A  =  { x } )
5 0ex 4531 . . . 4  |-  (/)  e.  _V
65elsnc2 4017 . . 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 1370    e. wcel 1758   E.wrex 2800    u. cun 3435   (/)c0 3746   {csn 3986  sngl bj-csngl 32791  tag bj-ctag 32800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-rex 2805  df-v 3080  df-dif 3440  df-un 3442  df-nul 3747  df-sn 3987  df-pr 3989  df-bj-sngl 32792  df-bj-tag 32801
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator