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Theorem rint0 3898
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 3862 . . 3  |-  ( X  =  (/)  ->  |^| X  =  |^| (/) )
21ineq2d 3351 . 2  |-  ( X  =  (/)  ->  ( A  i^i  |^| X )  =  ( A  i^i  |^| (/) ) )
3 int0 3873 . . . 4  |-  |^| (/)  =  _V
43ineq2i 3348 . . 3  |-  ( A  i^i  |^| (/) )  =  ( A  i^i  _V )
5 inv1 3474 . . 3  |-  ( A  i^i  _V )  =  A
64, 5eqtri 2210 . 2  |-  ( A  i^i  |^| (/) )  =  A
72, 6eqtrdi 2238 1  |-  ( X  =  (/)  ->  ( A  i^i  |^| X )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1364   _Vcvv 2752    i^i cin 3143   (/)c0 3437   |^|cint 3859
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-v 2754  df-dif 3146  df-in 3150  df-ss 3157  df-nul 3438  df-int 3860
This theorem is referenced by: (None)
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