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Theorem rint0 3918
Description: Relative intersection of an empty set. (Contributed by Stefan O'Rear, 3-Apr-2015.)
Assertion
Ref Expression
rint0  |-  ( X  =  (/)  ->  ( A  i^i  |^| X )  =  A )

Proof of Theorem rint0
StepHypRef Expression
1 inteq 3881 . . 3  |-  ( X  =  (/)  ->  |^| X  =  |^| (/) )
21ineq2d 3383 . 2  |-  ( X  =  (/)  ->  ( A  i^i  |^| X )  =  ( A  i^i  |^| (/) ) )
3 int0 3892 . . . 4  |-  |^| (/)  =  _V
43ineq2i 3380 . . 3  |-  ( A  i^i  |^| (/) )  =  ( A  i^i  _V )
5 inv1 3494 . . 3  |-  ( A  i^i  _V )  =  A
64, 5eqtri 2316 . 2  |-  ( A  i^i  |^| (/) )  =  A
72, 6syl6eq 2344 1  |-  ( X  =  (/)  ->  ( A  i^i  |^| X )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632   _Vcvv 2801    i^i cin 3164   (/)c0 3468   |^|cint 3878
This theorem is referenced by:  mrerintcl  13515  ismred2  13521  txtube  17350
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-v 2803  df-dif 3168  df-in 3172  df-ss 3179  df-nul 3469  df-int 3879
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