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Theorem rint0 4269
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 4232 . . 3  |-  ( X  =  (/)  ->  |^| X  =  |^| (/) )
21ineq2d 3653 . 2  |-  ( X  =  (/)  ->  ( A  i^i  |^| X )  =  ( A  i^i  |^| (/) ) )
3 int0 4243 . . . 4  |-  |^| (/)  =  _V
43ineq2i 3650 . . 3  |-  ( A  i^i  |^| (/) )  =  ( A  i^i  _V )
5 inv1 3765 . . 3  |-  ( A  i^i  _V )  =  A
64, 5eqtri 2480 . 2  |-  ( A  i^i  |^| (/) )  =  A
72, 6syl6eq 2508 1  |-  ( X  =  (/)  ->  ( A  i^i  |^| X )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370   _Vcvv 3071    i^i cin 3428   (/)c0 3738   |^|cint 4229
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 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ral 2800  df-v 3073  df-dif 3432  df-in 3436  df-ss 3443  df-nul 3739  df-int 4230
This theorem is referenced by:  incexclem  13410  incexc  13411  mrerintcl  14646  ismred2  14652  txtube  19338
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