ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  riotacl2 Unicode version

Theorem riotacl2 5533
Description: Membership law for "the unique element in  A such that  ph."

(Contributed by NM, 21-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)

Assertion
Ref Expression
riotacl2  |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A  ph )  e.  { x  e.  A  |  ph }
)

Proof of Theorem riotacl2
StepHypRef Expression
1 df-reu 2360 . . 3  |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
)
2 iotacl 4940 . . 3  |-  ( E! x ( x  e.  A  /\  ph )  ->  ( iota x ( x  e.  A  /\  ph ) )  e.  {
x  |  ( x  e.  A  /\  ph ) } )
31, 2sylbi 119 . 2  |-  ( E! x  e.  A  ph  ->  ( iota x ( x  e.  A  /\  ph ) )  e.  {
x  |  ( x  e.  A  /\  ph ) } )
4 df-riota 5520 . 2  |-  ( iota_ x  e.  A  ph )  =  ( iota x
( x  e.  A  /\  ph ) )
5 df-rab 2362 . 2  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
63, 4, 53eltr4g 2168 1  |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A  ph )  e.  { x  e.  A  |  ph }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    e. wcel 1434   E!weu 1943   {cab 2069   E!wreu 2355   {crab 2357   iotacio 4915   iota_crio 5519
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-un 2986  df-sn 3422  df-pr 3423  df-uni 3622  df-iota 4917  df-riota 5520
This theorem is referenced by:  riotacl  5534  riotasbc  5535  supubti  6507  suplubti  6508
  Copyright terms: Public domain W3C validator