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

Theorem rintm 3905
Description: Relative intersection of an inhabited class. (Contributed by Jim Kingdon, 19-Aug-2018.)
Assertion
Ref Expression
rintm  |-  ( ( X  C_  ~P A  /\  E. x  x  e.  X )  ->  ( A  i^i  |^| X )  = 
|^| X )
Distinct variable group:    x, X
Allowed substitution hint:    A( x)

Proof of Theorem rintm
StepHypRef Expression
1 incom 3268 . 2  |-  ( A  i^i  |^| X )  =  ( |^| X  i^i  A )
2 intssuni2m 3795 . . . 4  |-  ( ( X  C_  ~P A  /\  E. x  x  e.  X )  ->  |^| X  C_ 
U. ~P A )
3 ssid 3117 . . . . 5  |-  ~P A  C_ 
~P A
4 sspwuni 3897 . . . . 5  |-  ( ~P A  C_  ~P A  <->  U. ~P A  C_  A
)
53, 4mpbi 144 . . . 4  |-  U. ~P A  C_  A
62, 5sstrdi 3109 . . 3  |-  ( ( X  C_  ~P A  /\  E. x  x  e.  X )  ->  |^| X  C_  A )
7 df-ss 3084 . . 3  |-  ( |^| X  C_  A  <->  ( |^| X  i^i  A )  = 
|^| X )
86, 7sylib 121 . 2  |-  ( ( X  C_  ~P A  /\  E. x  x  e.  X )  ->  ( |^| X  i^i  A )  =  |^| X )
91, 8syl5eq 2184 1  |-  ( ( X  C_  ~P A  /\  E. x  x  e.  X )  ->  ( A  i^i  |^| X )  = 
|^| X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331   E.wex 1468    e. wcel 1480    i^i cin 3070    C_ wss 3071   ~Pcpw 3510   U.cuni 3736   |^|cint 3771
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-in 3077  df-ss 3084  df-pw 3512  df-uni 3737  df-int 3772
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator