Theorem sniota 5878

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 2480	. 2 |- F/ x E! x ph
2		nfab1 2766	. 2 |- F/_ x { x | ph }
3		nfiota1 5853	. . 3 |- F/_ x ( iota x ph )
4	3	nfsn 4242	. 2 |- F/_ x { ( iota x ph ) }
5		iota1 5865	. . . 4 |- ( E! x ph -> ( ph <-> ( iota x ph ) = x ) )
6		eqcom 2629	. . . 4 |- ( ( iota x ph ) = x <-> x = ( iota x ph ) )
7	5, 6	syl6bb 276	. . 3 |- ( E! x ph -> ( ph <-> x = ( iota x ph ) ) )
8		abid 2610	. . 3 |- ( x e. { x | ph } <-> ph )
9		velsn 4193	. . 3 |- ( x e. { ( iota x ph ) } <-> x = ( iota x ph ) )
10	7, 8, 9	3bitr4g 303	. 2 |- ( E! x ph -> ( x e. { x | ph } <-> x e. { ( iota x ph ) } ) )
11	1, 2, 4, 10	eqrd 3622	1 |- ( E! x ph -> { x | ph } = { ( iota x ph ) } )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   E!weu 2470   {cab 2608   {csn 4177   iotacio 5849

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-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-sbc 3436  df-un 3579  df-sn 4178  df-pr 4180  df-uni 4437  df-iota 5851

This theorem is referenced by:  snriota  6641