Theorem unisnif 32032

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

Step	Hyp	Ref	Expression
1		iftrue 4092	. . . 4 |- ( A e. _V -> if ( A e. _V , A , (/) ) = A )
2		unisng 4452	. . . 4 |- ( A e. _V -> U. { A } = A )
3	1, 2	eqtr4d 2659	. . 3 |- ( A e. _V -> if ( A e. _V , A , (/) ) = U. { A } )
4		iffalse 4095	. . . 4 |- ( -. A e. _V -> if ( A e. _V , A , (/) ) = (/) )
5		snprc 4253	. . . . . . 7 |- ( -. A e. _V <-> { A } = (/) )
6	5	biimpi 206	. . . . . 6 |- ( -. A e. _V -> { A } = (/) )
7	6	unieqd 4446	. . . . 5 |- ( -. A e. _V -> U. { A } = U. (/) )
8		uni0 4465	. . . . 5 |- U. (/) = (/)
9	7, 8	syl6eq 2672	. . . 4 |- ( -. A e. _V -> U. { A } = (/) )
10	4, 9	eqtr4d 2659	. . 3 |- ( -. A e. _V -> if ( A e. _V , A , (/) ) = U. { A } )
11	3, 10	pm2.61i 176	. 2 |- if ( A e. _V , A , (/) ) = U. { A }
12	11	eqcomi 2631	1 |- U. { A } = if ( A e. _V , A , (/) )

Syntax hints:   -. wn 3   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   ifcif 4086   {csn 4177   U.cuni 4436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-uni 4437

This theorem is referenced by:  dfrdg4  32058