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Theorem rintn0 4371
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 3624 . 2  |-  ( A  i^i  |^| X )  =  ( |^| X  i^i  A )
2 intssuni2 4259 . . . 4  |-  ( ( X  C_  ~P A  /\  X  =/=  (/) )  ->  |^| X  C_  U. ~P A
)
3 ssid 3450 . . . . 5  |-  ~P A  C_ 
~P A
4 sspwuni 4366 . . . . 5  |-  ( ~P A  C_  ~P A  <->  U. ~P A  C_  A
)
53, 4mpbi 212 . . . 4  |-  U. ~P A  C_  A
62, 5syl6ss 3443 . . 3  |-  ( ( X  C_  ~P A  /\  X  =/=  (/) )  ->  |^| X  C_  A )
7 df-ss 3417 . . 3  |-  ( |^| X  C_  A  <->  ( |^| X  i^i  A )  = 
|^| X )
86, 7sylib 200 . 2  |-  ( ( X  C_  ~P A  /\  X  =/=  (/) )  -> 
( |^| X  i^i  A
)  =  |^| X
)
91, 8syl5eq 2496 1  |-  ( ( X  C_  ~P A  /\  X  =/=  (/) )  -> 
( A  i^i  |^| X )  =  |^| X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1443    =/= wne 2621    i^i cin 3402    C_ wss 3403   (/)c0 3730   ~Pcpw 3950   U.cuni 4197   |^|cint 4233
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-v 3046  df-dif 3406  df-in 3410  df-ss 3417  df-nul 3731  df-pw 3952  df-uni 4198  df-int 4234
This theorem is referenced by:  mrerintcl  15496  ismred2  15502
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