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Theorem unisnif 29768
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 3880 . . . 4  |-  ( A  e.  _V  ->  if ( A  e.  _V ,  A ,  (/) )  =  A )
2 unisng 4196 . . . 4  |-  ( A  e.  _V  ->  U. { A }  =  A
)
31, 2eqtr4d 2440 . . 3  |-  ( A  e.  _V  ->  if ( A  e.  _V ,  A ,  (/) )  = 
U. { A }
)
4 iffalse 3883 . . . 4  |-  ( -.  A  e.  _V  ->  if ( A  e.  _V ,  A ,  (/) )  =  (/) )
5 snprc 4024 . . . . . . 7  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
65biimpi 194 . . . . . 6  |-  ( -.  A  e.  _V  ->  { A }  =  (/) )
76unieqd 4190 . . . . 5  |-  ( -.  A  e.  _V  ->  U. { A }  =  U. (/) )
8 uni0 4207 . . . . 5  |-  U. (/)  =  (/)
97, 8syl6eq 2453 . . . 4  |-  ( -.  A  e.  _V  ->  U. { A }  =  (/) )
104, 9eqtr4d 2440 . . 3  |-  ( -.  A  e.  _V  ->  if ( A  e.  _V ,  A ,  (/) )  = 
U. { A }
)
113, 10pm2.61i 164 . 2  |-  if ( A  e.  _V ,  A ,  (/) )  = 
U. { A }
1211eqcomi 2409 1  |-  U. { A }  =  if ( A  e.  _V ,  A ,  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1399    e. wcel 1836   _Vcvv 3051   (/)c0 3728   ifcif 3874   {csn 3961   U.cuni 4180
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-ral 2751  df-rex 2752  df-v 3053  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3729  df-if 3875  df-sn 3962  df-pr 3964  df-uni 4181
This theorem is referenced by:  dfrdg4  29793
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