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Theorem unisnif 24464
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 3571 . . . 4  |-  ( A  e.  _V  ->  if ( A  e.  _V ,  A ,  (/) )  =  A )
2 unisng 3844 . . . 4  |-  ( A  e.  _V  ->  U. { A }  =  A
)
31, 2eqtr4d 2318 . . 3  |-  ( A  e.  _V  ->  if ( A  e.  _V ,  A ,  (/) )  = 
U. { A }
)
4 iffalse 3572 . . . 4  |-  ( -.  A  e.  _V  ->  if ( A  e.  _V ,  A ,  (/) )  =  (/) )
5 snprc 3695 . . . . . . 7  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
65biimpi 186 . . . . . 6  |-  ( -.  A  e.  _V  ->  { A }  =  (/) )
76unieqd 3838 . . . . 5  |-  ( -.  A  e.  _V  ->  U. { A }  =  U. (/) )
8 uni0 3854 . . . . 5  |-  U. (/)  =  (/)
97, 8syl6eq 2331 . . . 4  |-  ( -.  A  e.  _V  ->  U. { A }  =  (/) )
104, 9eqtr4d 2318 . . 3  |-  ( -.  A  e.  _V  ->  if ( A  e.  _V ,  A ,  (/) )  = 
U. { A }
)
113, 10pm2.61i 156 . 2  |-  if ( A  e.  _V ,  A ,  (/) )  = 
U. { A }
1211eqcomi 2287 1  |-  U. { A }  =  if ( A  e.  _V ,  A ,  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1623    e. wcel 1684   _Vcvv 2788   (/)c0 3455   ifcif 3565   {csn 3640   U.cuni 3827
This theorem is referenced by:  dfrdg4  24488
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-uni 3828
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