MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sniota Unicode version

Theorem sniota 5408
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
StepHypRef Expression
1 nfeu1 2268 . . 3  |-  F/ x E! x ph
2 iota1 5395 . . . . 5  |-  ( E! x ph  ->  ( ph 
<->  ( iota x ph )  =  x )
)
3 eqcom 2410 . . . . 5  |-  ( ( iota x ph )  =  x  <->  x  =  ( iota x ph ) )
42, 3syl6bb 253 . . . 4  |-  ( E! x ph  ->  ( ph 
<->  x  =  ( iota
x ph ) ) )
5 abid 2396 . . . 4  |-  ( x  e.  { x  | 
ph }  <->  ph )
6 vex 2923 . . . . 5  |-  x  e. 
_V
76elsnc 3801 . . . 4  |-  ( x  e.  { ( iota
x ph ) }  <->  x  =  ( iota x ph )
)
84, 5, 73bitr4g 280 . . 3  |-  ( E! x ph  ->  (
x  e.  { x  |  ph }  <->  x  e.  { ( iota x ph ) } ) )
91, 8alrimi 1777 . 2  |-  ( E! x ph  ->  A. x
( x  e.  {
x  |  ph }  <->  x  e.  { ( iota
x ph ) } ) )
10 nfab1 2546 . . 3  |-  F/_ x { x  |  ph }
11 nfiota1 5383 . . . 4  |-  F/_ x
( iota x ph )
1211nfsn 3830 . . 3  |-  F/_ x { ( iota x ph ) }
1310, 12cleqf 2568 . 2  |-  ( { x  |  ph }  =  { ( iota x ph ) }  <->  A. x
( x  e.  {
x  |  ph }  <->  x  e.  { ( iota
x ph ) } ) )
149, 13sylibr 204 1  |-  ( E! x ph  ->  { x  |  ph }  =  {
( iota x ph ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1546    = wceq 1649    e. wcel 1721   E!weu 2258   {cab 2394   {csn 3778   iotacio 5379
This theorem is referenced by:  snriota  6543
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ral 2675  df-rex 2676  df-v 2922  df-sbc 3126  df-un 3289  df-sn 3784  df-pr 3785  df-uni 3980  df-iota 5381
  Copyright terms: Public domain W3C validator