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Theorem rint0 4090
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 4053 . . 3  |-  ( X  =  (/)  ->  |^| X  =  |^| (/) )
21ineq2d 3542 . 2  |-  ( X  =  (/)  ->  ( A  i^i  |^| X )  =  ( A  i^i  |^| (/) ) )
3 int0 4064 . . . 4  |-  |^| (/)  =  _V
43ineq2i 3539 . . 3  |-  ( A  i^i  |^| (/) )  =  ( A  i^i  _V )
5 inv1 3654 . . 3  |-  ( A  i^i  _V )  =  A
64, 5eqtri 2456 . 2  |-  ( A  i^i  |^| (/) )  =  A
72, 6syl6eq 2484 1  |-  ( X  =  (/)  ->  ( A  i^i  |^| X )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652   _Vcvv 2956    i^i cin 3319   (/)c0 3628   |^|cint 4050
This theorem is referenced by:  incexclem  12616  incexc  12617  mrerintcl  13822  ismred2  13828  txtube  17672
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-v 2958  df-dif 3323  df-in 3327  df-ss 3334  df-nul 3629  df-int 4051
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