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Theorem sniota 4922
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 1927 . . 3  |-  F/ x E! x ph
2 iota1 4909 . . . . 5  |-  ( E! x ph  ->  ( ph 
<->  ( iota x ph )  =  x )
)
3 eqcom 2058 . . . . 5  |-  ( ( iota x ph )  =  x  <->  x  =  ( iota x ph ) )
42, 3syl6bb 189 . . . 4  |-  ( E! x ph  ->  ( ph 
<->  x  =  ( iota
x ph ) ) )
5 abid 2044 . . . 4  |-  ( x  e.  { x  | 
ph }  <->  ph )
6 vex 2577 . . . . 5  |-  x  e. 
_V
76elsn 3419 . . . 4  |-  ( x  e.  { ( iota
x ph ) }  <->  x  =  ( iota x ph )
)
84, 5, 73bitr4g 216 . . 3  |-  ( E! x ph  ->  (
x  e.  { x  |  ph }  <->  x  e.  { ( iota x ph ) } ) )
91, 8alrimi 1431 . 2  |-  ( E! x ph  ->  A. x
( x  e.  {
x  |  ph }  <->  x  e.  { ( iota
x ph ) } ) )
10 nfab1 2196 . . 3  |-  F/_ x { x  |  ph }
11 nfiota1 4897 . . . 4  |-  F/_ x
( iota x ph )
1211nfsn 3458 . . 3  |-  F/_ x { ( iota x ph ) }
1310, 12cleqf 2217 . 2  |-  ( { x  |  ph }  =  { ( iota x ph ) }  <->  A. x
( x  e.  {
x  |  ph }  <->  x  e.  { ( iota
x ph ) } ) )
149, 13sylibr 141 1  |-  ( E! x ph  ->  { x  |  ph }  =  {
( iota x ph ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 102   A.wal 1257    = wceq 1259    e. wcel 1409   E!weu 1916   {cab 2042   {csn 3403   iotacio 4893
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-v 2576  df-sbc 2788  df-un 2950  df-sn 3409  df-pr 3410  df-uni 3609  df-iota 4895
This theorem is referenced by:  snriota  5525
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