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Theorem bj-sngltag 32572
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 32571 . 2  |-  ( { A }  e. sngl  B  ->  { A }  e. tag  B )
2 df-bj-tag 32564 . . . 4  |- tag  B  =  (sngl  B  u.  { (/)
} )
32eleq2i 2507 . . 3  |-  ( { A }  e. tag  B  <->  { A }  e.  (sngl 
B  u.  { (/) } ) )
4 elun 3518 . . . 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 3923 . . . . . 6  |-  ( { A }  e.  { (/)
}  ->  { A }  =  (/) )
7 snprc 3960 . . . . . . 7  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
8 elex 3002 . . . . . . . 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 1369    e. wcel 1756   _Vcvv 2993    u. cun 3347   (/)c0 3658   {csn 3898  sngl bj-csngl 32554  tag bj-ctag 32563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-v 2995  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-sn 3899  df-bj-tag 32564
This theorem is referenced by:  bj-tagcg  32574  bj-taginv  32575
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