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Theorem unisnif 25770
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 3745 . . . 4  |-  ( A  e.  _V  ->  if ( A  e.  _V ,  A ,  (/) )  =  A )
2 unisng 4032 . . . 4  |-  ( A  e.  _V  ->  U. { A }  =  A
)
31, 2eqtr4d 2471 . . 3  |-  ( A  e.  _V  ->  if ( A  e.  _V ,  A ,  (/) )  = 
U. { A }
)
4 iffalse 3746 . . . 4  |-  ( -.  A  e.  _V  ->  if ( A  e.  _V ,  A ,  (/) )  =  (/) )
5 snprc 3871 . . . . . . 7  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
65biimpi 187 . . . . . 6  |-  ( -.  A  e.  _V  ->  { A }  =  (/) )
76unieqd 4026 . . . . 5  |-  ( -.  A  e.  _V  ->  U. { A }  =  U. (/) )
8 uni0 4042 . . . . 5  |-  U. (/)  =  (/)
97, 8syl6eq 2484 . . . 4  |-  ( -.  A  e.  _V  ->  U. { A }  =  (/) )
104, 9eqtr4d 2471 . . 3  |-  ( -.  A  e.  _V  ->  if ( A  e.  _V ,  A ,  (/) )  = 
U. { A }
)
113, 10pm2.61i 158 . 2  |-  if ( A  e.  _V ,  A ,  (/) )  = 
U. { A }
1211eqcomi 2440 1  |-  U. { A }  =  if ( A  e.  _V ,  A ,  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1652    e. wcel 1725   _Vcvv 2956   (/)c0 3628   ifcif 3739   {csn 3814   U.cuni 4015
This theorem is referenced by:  dfrdg4  25795
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-uni 4016
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