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Theorem rint0 4274
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 4236 . . 3  |-  ( X  =  (/)  ->  |^| X  =  |^| (/) )
21ineq2d 3633 . 2  |-  ( X  =  (/)  ->  ( A  i^i  |^| X )  =  ( A  i^i  |^| (/) ) )
3 int0 4247 . . . 4  |-  |^| (/)  =  _V
43ineq2i 3630 . . 3  |-  ( A  i^i  |^| (/) )  =  ( A  i^i  _V )
5 inv1 3760 . . 3  |-  ( A  i^i  _V )  =  A
64, 5eqtri 2472 . 2  |-  ( A  i^i  |^| (/) )  =  A
72, 6syl6eq 2500 1  |-  ( X  =  (/)  ->  ( A  i^i  |^| X )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1443   _Vcvv 3044    i^i cin 3402   (/)c0 3730   |^|cint 4233
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430
This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ral 2741  df-v 3046  df-dif 3406  df-in 3410  df-ss 3417  df-nul 3731  df-int 4234
This theorem is referenced by:  incexclem  13887  incexc  13888  mrerintcl  15496  ismred2  15502  txtube  20648
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