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Theorem rint0 4294
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 4256 . . 3  |-  ( X  =  (/)  ->  |^| X  =  |^| (/) )
21ineq2d 3665 . 2  |-  ( X  =  (/)  ->  ( A  i^i  |^| X )  =  ( A  i^i  |^| (/) ) )
3 int0 4267 . . . 4  |-  |^| (/)  =  _V
43ineq2i 3662 . . 3  |-  ( A  i^i  |^| (/) )  =  ( A  i^i  _V )
5 inv1 3790 . . 3  |-  ( A  i^i  _V )  =  A
64, 5eqtri 2452 . 2  |-  ( A  i^i  |^| (/) )  =  A
72, 6syl6eq 2480 1  |-  ( X  =  (/)  ->  ( A  i^i  |^| X )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1438   _Vcvv 3082    i^i cin 3436   (/)c0 3762   |^|cint 4253
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401
This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ral 2781  df-v 3084  df-dif 3440  df-in 3444  df-ss 3451  df-nul 3763  df-int 4254
This theorem is referenced by:  incexclem  13887  incexc  13888  mrerintcl  15496  ismred2  15502  txtube  20647
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