Theorem sniota 5343

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 2091	. . 3 |- F/ x E! x ph
2		iota1 5327	. . . . 5 |- ( E! x ph -> ( ph <-> ( iota x ph ) = x ) )
3		eqcom 2234	. . . . 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 2220	. . . 4 |- ( x e. { x | ph } <-> ph )
6		vex 2816	. . . . 5 |- x e. _V
7	6	elsn 3705	. . . 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 1571	. 2 |- ( E! x ph -> A. x ( x e. { x | ph } <-> x e. { ( iota x ph ) } ) )
10		nfab1 2386	. . 3 |- F/_ x { x | ph }
11		nfiota1 5314	. . . 4 |- F/_ x ( iota x ph )
12	11	nfsn 3749	. . 3 |- F/_ x { ( iota x ph ) }
13	10, 12	cleqf 2409	. 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 1396   = wceq 1398   E!weu 2080   e. wcel 2203   {cab 2218   {csn 3689   iotacio 5310

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-rex 2526  df-v 2815  df-sbc 3043  df-un 3215  df-sn 3695  df-pr 3696  df-uni 3915  df-iota 5312

This theorem is referenced by:  snriota  6035