Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-0eltag Structured version   Unicode version

Theorem bj-0eltag 32804
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
StepHypRef Expression
1 0ex 4531 . . . . 5  |-  (/)  e.  _V
21snid 4014 . . . 4  |-  (/)  e.  { (/)
}
32olci 391 . . 3  |-  ( (/)  e. sngl  A  \/  (/)  e.  { (/)
} )
4 elun 3606 . . 3  |-  ( (/)  e.  (sngl  A  u.  { (/)
} )  <->  ( (/)  e. sngl  A  \/  (/)  e.  { (/) } ) )
53, 4mpbir 209 . 2  |-  (/)  e.  (sngl 
A  u.  { (/) } )
6 df-bj-tag 32801 . 2  |- tag  A  =  (sngl  A  u.  { (/)
} )
75, 6eleqtrri 2541 1  |-  (/)  e. tag  A
Colors of variables: wff setvar class
Syntax hints:    \/ wo 368    e. wcel 1758    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-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-nul 4530
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-v 3080  df-dif 3440  df-un 3442  df-nul 3747  df-sn 3987  df-bj-tag 32801
This theorem is referenced by:  bj-tagn0  32805
  Copyright terms: Public domain W3C validator