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Theorem unisnif 24535
Description: Express union of singleton in terms of  if. (Contributed by Scott Fenton, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
unisnif  |-  U. { A }  =  if ( A  e.  _V ,  A ,  (/) )

Proof of Theorem unisnif
StepHypRef Expression
1 iftrue 3584 . . . 4  |-  ( A  e.  _V  ->  if ( A  e.  _V ,  A ,  (/) )  =  A )
2 unisng 3860 . . . 4  |-  ( A  e.  _V  ->  U. { A }  =  A
)
31, 2eqtr4d 2331 . . 3  |-  ( A  e.  _V  ->  if ( A  e.  _V ,  A ,  (/) )  = 
U. { A }
)
4 iffalse 3585 . . . 4  |-  ( -.  A  e.  _V  ->  if ( A  e.  _V ,  A ,  (/) )  =  (/) )
5 snprc 3708 . . . . . . 7  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
65biimpi 186 . . . . . 6  |-  ( -.  A  e.  _V  ->  { A }  =  (/) )
76unieqd 3854 . . . . 5  |-  ( -.  A  e.  _V  ->  U. { A }  =  U. (/) )
8 uni0 3870 . . . . 5  |-  U. (/)  =  (/)
97, 8syl6eq 2344 . . . 4  |-  ( -.  A  e.  _V  ->  U. { A }  =  (/) )
104, 9eqtr4d 2331 . . 3  |-  ( -.  A  e.  _V  ->  if ( A  e.  _V ,  A ,  (/) )  = 
U. { A }
)
113, 10pm2.61i 156 . 2  |-  if ( A  e.  _V ,  A ,  (/) )  = 
U. { A }
1211eqcomi 2300 1  |-  U. { A }  =  if ( A  e.  _V ,  A ,  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1632    e. wcel 1696   _Vcvv 2801   (/)c0 3468   ifcif 3578   {csn 3653   U.cuni 3843
This theorem is referenced by:  dfrdg4  24560
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-uni 3844
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