Theorem sniota 5249

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 2056	. . 3 |- F/ x E! x ph
2		iota1 5233	. . . . 5 |- ( E! x ph -> ( ph <-> ( iota x ph ) = x ) )
3		eqcom 2198	. . . . 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 2184	. . . 4 |- ( x e. { x | ph } <-> ph )
6		vex 2766	. . . . 5 |- x e. _V
7	6	elsn 3638	. . . 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 1536	. 2 |- ( E! x ph -> A. x ( x e. { x | ph } <-> x e. { ( iota x ph ) } ) )
10		nfab1 2341	. . 3 |- F/_ x { x | ph }
11		nfiota1 5221	. . . 4 |- F/_ x ( iota x ph )
12	11	nfsn 3682	. . 3 |- F/_ x { ( iota x ph ) }
13	10, 12	cleqf 2364	. 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 2045   e. wcel 2167   {cab 2182   {csn 3622   iotacio 5217

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-sn 3628  df-pr 3629  df-uni 3840  df-iota 5219

This theorem is referenced by:  snriota  5907