Theorem sniota 5281

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 2066	. . 3 |- F/ x E! x ph
2		iota1 5265	. . . . 5 |- ( E! x ph -> ( ph <-> ( iota x ph ) = x ) )
3		eqcom 2209	. . . . 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 2195	. . . 4 |- ( x e. { x | ph } <-> ph )
6		vex 2779	. . . . 5 |- x e. _V
7	6	elsn 3659	. . . 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 1546	. 2 |- ( E! x ph -> A. x ( x e. { x | ph } <-> x e. { ( iota x ph ) } ) )
10		nfab1 2352	. . 3 |- F/_ x { x | ph }
11		nfiota1 5253	. . . 4 |- F/_ x ( iota x ph )
12	11	nfsn 3703	. . 3 |- F/_ x { ( iota x ph ) }
13	10, 12	cleqf 2375	. 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 1371   = wceq 1373   E!weu 2055   e. wcel 2178   {cab 2193   {csn 3643   iotacio 5249

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rex 2492  df-v 2778  df-sbc 3006  df-un 3178  df-sn 3649  df-pr 3650  df-uni 3865  df-iota 5251

This theorem is referenced by:  snriota  5952