MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rint0 Structured version   Unicode version

Theorem rint0 4270
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 3643 . 2  |-  ( X  =  (/)  ->  ( A  i^i  |^| X )  =  ( A  i^i  |^| (/) ) )
3 int0 4243 . . . 4  |-  |^| (/)  =  _V
43ineq2i 3640 . . 3  |-  ( A  i^i  |^| (/) )  =  ( A  i^i  _V )
5 inv1 3768 . . 3  |-  ( A  i^i  _V )  =  A
64, 5eqtri 2433 . 2  |-  ( A  i^i  |^| (/) )  =  A
72, 6syl6eq 2461 1  |-  ( X  =  (/)  ->  ( A  i^i  |^| X )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1407   _Vcvv 3061    i^i cin 3415   (/)c0 3740   |^|cint 4229
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382
This theorem depends on definitions:  df-bi 187  df-an 371  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ral 2761  df-v 3063  df-dif 3419  df-in 3423  df-ss 3430  df-nul 3741  df-int 4230
This theorem is referenced by:  incexclem  13801  incexc  13802  mrerintcl  15213  ismred2  15219  txtube  20435
  Copyright terms: Public domain W3C validator