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Theorem rintn0 4422
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 3696 . 2  |-  ( A  i^i  |^| X )  =  ( |^| X  i^i  A )
2 intssuni2 4313 . . . 4  |-  ( ( X  C_  ~P A  /\  X  =/=  (/) )  ->  |^| X  C_  U. ~P A
)
3 ssid 3528 . . . . 5  |-  ~P A  C_ 
~P A
4 sspwuni 4417 . . . . 5  |-  ( ~P A  C_  ~P A  <->  U. ~P A  C_  A
)
53, 4mpbi 208 . . . 4  |-  U. ~P A  C_  A
62, 5syl6ss 3521 . . 3  |-  ( ( X  C_  ~P A  /\  X  =/=  (/) )  ->  |^| X  C_  A )
7 df-ss 3495 . . 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 2520 1  |-  ( ( X  C_  ~P A  /\  X  =/=  (/) )  -> 
( A  i^i  |^| X )  =  |^| X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    =/= wne 2662    i^i cin 3480    C_ wss 3481   (/)c0 3790   ~Pcpw 4016   U.cuni 4251   |^|cint 4288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-v 3120  df-dif 3484  df-in 3488  df-ss 3495  df-nul 3791  df-pw 4018  df-uni 4252  df-int 4289
This theorem is referenced by:  mrerintcl  14869  ismred2  14875
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