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Theorem rintn0 4370
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 3652 . 2  |-  ( A  i^i  |^| X )  =  ( |^| X  i^i  A )
2 intssuni2 4262 . . . 4  |-  ( ( X  C_  ~P A  /\  X  =/=  (/) )  ->  |^| X  C_  U. ~P A
)
3 ssid 3484 . . . . 5  |-  ~P A  C_ 
~P A
4 sspwuni 4365 . . . . 5  |-  ( ~P A  C_  ~P A  <->  U. ~P A  C_  A
)
53, 4mpbi 208 . . . 4  |-  U. ~P A  C_  A
62, 5syl6ss 3477 . . 3  |-  ( ( X  C_  ~P A  /\  X  =/=  (/) )  ->  |^| X  C_  A )
7 df-ss 3451 . . 3  |-  ( |^| X  C_  A  <->  ( |^| X  i^i  A )  = 
|^| X )
86, 7sylib 196 . 2  |-  ( ( X  C_  ~P A  /\  X  =/=  (/) )  -> 
( |^| X  i^i  A
)  =  |^| X
)
91, 8syl5eq 2507 1  |-  ( ( X  C_  ~P A  /\  X  =/=  (/) )  -> 
( A  i^i  |^| X )  =  |^| X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    =/= wne 2648    i^i cin 3436    C_ wss 3437   (/)c0 3746   ~Pcpw 3969   U.cuni 4200   |^|cint 4237
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-v 3080  df-dif 3440  df-in 3444  df-ss 3451  df-nul 3747  df-pw 3971  df-uni 4201  df-int 4238
This theorem is referenced by:  mrerintcl  14655  ismred2  14661
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