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Theorem bj-sngltag 33839
Description: The singletonization and the tagging of a set contain the same singletons. (Contributed by BJ, 6-Oct-2018.)
Assertion
Ref Expression
bj-sngltag  |-  ( A  e.  V  ->  ( { A }  e. sngl  B  <->  { A }  e. tag  B
) )

Proof of Theorem bj-sngltag
StepHypRef Expression
1 bj-sngltagi 33838 . 2  |-  ( { A }  e. sngl  B  ->  { A }  e. tag  B )
2 df-bj-tag 33831 . . . 4  |- tag  B  =  (sngl  B  u.  { (/)
} )
32eleq2i 2545 . . 3  |-  ( { A }  e. tag  B  <->  { A }  e.  (sngl 
B  u.  { (/) } ) )
4 elun 3645 . . . 4  |-  ( { A }  e.  (sngl 
B  u.  { (/) } )  <->  ( { A }  e. sngl  B  \/  { A }  e.  { (/) } ) )
5 idd 24 . . . . 5  |-  ( A  e.  V  ->  ( { A }  e. sngl  B  ->  { A }  e. sngl  B ) )
6 elsni 4052 . . . . . 6  |-  ( { A }  e.  { (/)
}  ->  { A }  =  (/) )
7 snprc 4091 . . . . . . 7  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
8 elex 3122 . . . . . . . 8  |-  ( A  e.  V  ->  A  e.  _V )
98pm2.24d 143 . . . . . . 7  |-  ( A  e.  V  ->  ( -.  A  e.  _V  ->  { A }  e. sngl  B ) )
107, 9syl5bir 218 . . . . . 6  |-  ( A  e.  V  ->  ( { A }  =  (/)  ->  { A }  e. sngl  B ) )
116, 10syl5 32 . . . . 5  |-  ( A  e.  V  ->  ( { A }  e.  { (/)
}  ->  { A }  e. sngl  B )
)
125, 11jaod 380 . . . 4  |-  ( A  e.  V  ->  (
( { A }  e. sngl  B  \/  { A }  e.  { (/) } )  ->  { A }  e. sngl  B ) )
134, 12syl5bi 217 . . 3  |-  ( A  e.  V  ->  ( { A }  e.  (sngl 
B  u.  { (/) } )  ->  { A }  e. sngl  B )
)
143, 13syl5bi 217 . 2  |-  ( A  e.  V  ->  ( { A }  e. tag  B  ->  { A }  e. sngl  B ) )
151, 14impbid2 204 1  |-  ( A  e.  V  ->  ( { A }  e. sngl  B  <->  { A }  e. tag  B
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    = wceq 1379    e. wcel 1767   _Vcvv 3113    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-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
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-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-sn 4028  df-bj-tag 33831
This theorem is referenced by:  bj-tagcg  33841  bj-taginv  33842
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