MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rintn0 Unicode version

Theorem rintn0 4073
Description: Relative intersection of a nonempty set. (Contributed by Stefan O'Rear, 3-Apr-2015.) (Revised by Mario Carneiro, 5-Jun-2015.)
Assertion
Ref Expression
rintn0  |-  ( ( X  C_  ~P A  /\  X  =/=  (/) )  -> 
( A  i^i  |^| X )  =  |^| X )

Proof of Theorem rintn0
StepHypRef Expression
1 incom 3437 . 2  |-  ( A  i^i  |^| X )  =  ( |^| X  i^i  A )
2 intssuni2 3968 . . . 4  |-  ( ( X  C_  ~P A  /\  X  =/=  (/) )  ->  |^| X  C_  U. ~P A
)
3 ssid 3273 . . . . 5  |-  ~P A  C_ 
~P A
4 sspwuni 4068 . . . . 5  |-  ( ~P A  C_  ~P A  <->  U. ~P A  C_  A
)
53, 4mpbi 199 . . . 4  |-  U. ~P A  C_  A
62, 5syl6ss 3267 . . 3  |-  ( ( X  C_  ~P A  /\  X  =/=  (/) )  ->  |^| X  C_  A )
7 df-ss 3242 . . 3  |-  ( |^| X  C_  A  <->  ( |^| X  i^i  A )  = 
|^| X )
86, 7sylib 188 . 2  |-  ( ( X  C_  ~P A  /\  X  =/=  (/) )  -> 
( |^| X  i^i  A
)  =  |^| X
)
91, 8syl5eq 2402 1  |-  ( ( X  C_  ~P A  /\  X  =/=  (/) )  -> 
( A  i^i  |^| X )  =  |^| X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    =/= wne 2521    i^i cin 3227    C_ wss 3228   (/)c0 3531   ~Pcpw 3701   U.cuni 3908   |^|cint 3943
This theorem is referenced by:  mrerintcl  13598  ismred2  13604
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-v 2866  df-dif 3231  df-in 3235  df-ss 3242  df-nul 3532  df-pw 3703  df-uni 3909  df-int 3944
  Copyright terms: Public domain W3C validator