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Theorem rintn0 3992
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 3361 . 2  |-  ( A  i^i  |^| X )  =  ( |^| X  i^i  A )
2 intssuni2 3887 . . . 4  |-  ( ( X  C_  ~P A  /\  X  =/=  (/) )  ->  |^| X  C_  U. ~P A
)
3 ssid 3197 . . . . 5  |-  ~P A  C_ 
~P A
4 sspwuni 3987 . . . . 5  |-  ( ~P A  C_  ~P A  <->  U. ~P A  C_  A
)
53, 4mpbi 199 . . . 4  |-  U. ~P A  C_  A
62, 5syl6ss 3191 . . 3  |-  ( ( X  C_  ~P A  /\  X  =/=  (/) )  ->  |^| X  C_  A )
7 df-ss 3166 . . 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 2327 1  |-  ( ( X  C_  ~P A  /\  X  =/=  (/) )  -> 
( A  i^i  |^| X )  =  |^| X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    =/= wne 2446    i^i cin 3151    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   U.cuni 3827   |^|cint 3862
This theorem is referenced by:  mrerintcl  13499  ismred2  13505
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-v 2790  df-dif 3155  df-in 3159  df-ss 3166  df-nul 3456  df-pw 3627  df-uni 3828  df-int 3863
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