Theorem sniota 5229

Description: A class abstraction with a unique member can be expressed as a singleton. (Contributed by Mario Carneiro, 23-Dec-2016.)

Assertion

Ref	Expression
sniota	|- ( E! x ph -> { x | ph } = { ( iota x ph ) } )

Proof of Theorem sniota

Step	Hyp	Ref	Expression
1		nfeu1 2049	. . 3 |- F/ x E! x ph
2		iota1 5213	. . . . 5 |- ( E! x ph -> ( ph <-> ( iota x ph ) = x ) )
3		eqcom 2191	. . . . 5 |- ( ( iota x ph ) = x <-> x = ( iota x ph ) )
4	2, 3	bitrdi 196	. . . 4 |- ( E! x ph -> ( ph <-> x = ( iota x ph ) ) )
5		abid 2177	. . . 4 |- ( x e. { x | ph } <-> ph )
6		vex 2755	. . . . 5 |- x e. _V
7	6	elsn 3626	. . . 4 |- ( x e. { ( iota x ph ) } <-> x = ( iota x ph ) )
8	4, 5, 7	3bitr4g 223	. . 3 |- ( E! x ph -> ( x e. { x | ph } <-> x e. { ( iota x ph ) } ) )
9	1, 8	alrimi 1533	. 2 |- ( E! x ph -> A. x ( x e. { x | ph } <-> x e. { ( iota x ph ) } ) )
10		nfab1 2334	. . 3 |- F/_ x { x | ph }
11		nfiota1 5201	. . . 4 |- F/_ x ( iota x ph )
12	11	nfsn 3670	. . 3 |- F/_ x { ( iota x ph ) }
13	10, 12	cleqf 2357	. 2 |- ( { x | ph } = { ( iota x ph ) } <-> A. x ( x e. { x | ph } <-> x e. { ( iota x ph ) } ) )
14	9, 13	sylibr 134	1 |- ( E! x ph -> { x | ph } = { ( iota x ph ) } )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1362   = wceq 1364   E!weu 2038   e. wcel 2160   {cab 2175   {csn 3610   iotacio 5197

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-sn 3616  df-pr 3617  df-uni 3828  df-iota 5199

This theorem is referenced by:  snriota  5885