Theorem sniota 4914

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 1952	. . 3 |- F/ x E! x ph
2		iota1 4901	. . . . 5 |- ( E! x ph -> ( ph <-> ( iota x ph ) = x ) )
3		eqcom 2083	. . . . 5 |- ( ( iota x ph ) = x <-> x = ( iota x ph ) )
4	2, 3	syl6bb 194	. . . 4 |- ( E! x ph -> ( ph <-> x = ( iota x ph ) ) )
5		abid 2069	. . . 4 |- ( x e. { x | ph } <-> ph )
6		vex 2604	. . . . 5 |- x e. _V
7	6	elsn 3414	. . . 4 |- ( x e. { ( iota x ph ) } <-> x = ( iota x ph ) )
8	4, 5, 7	3bitr4g 221	. . 3 |- ( E! x ph -> ( x e. { x | ph } <-> x e. { ( iota x ph ) } ) )
9	1, 8	alrimi 1455	. 2 |- ( E! x ph -> A. x ( x e. { x | ph } <-> x e. { ( iota x ph ) } ) )
10		nfab1 2221	. . 3 |- F/_ x { x | ph }
11		nfiota1 4889	. . . 4 |- F/_ x ( iota x ph )
12	11	nfsn 3452	. . 3 |- F/_ x { ( iota x ph ) }
13	10, 12	cleqf 2242	. 2 |- ( { x | ph } = { ( iota x ph ) } <-> A. x ( x e. { x | ph } <-> x e. { ( iota x ph ) } ) )
14	9, 13	sylibr 132	1 |- ( E! x ph -> { x | ph } = { ( iota x ph ) } )

Syntax hints:   -> wi 4   <-> wb 103   A.wal 1282   = wceq 1284   e. wcel 1433   E!weu 1941   {cab 2067   {csn 3398   iotacio 4885

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-sn 3404  df-pr 3405  df-uni 3602  df-iota 4887

This theorem is referenced by:  snriota  5517