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Theorem intprg 3897
Description: The intersection of a pair is the intersection of its members. Closed form of intpr 3896. Theorem 71 of [Suppes] p. 42. (Contributed by FL, 27-Apr-2008.)
Assertion
Ref Expression
intprg  |-  ( ( A  e.  V  /\  B  e.  W )  ->  |^| { A ,  B }  =  ( A  i^i  B ) )

Proof of Theorem intprg
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 preq1 3707 . . . 4  |-  ( x  =  A  ->  { x ,  y }  =  { A ,  y } )
21inteqd 3868 . . 3  |-  ( x  =  A  ->  |^| { x ,  y }  =  |^| { A ,  y } )
3 ineq1 3364 . . 3  |-  ( x  =  A  ->  (
x  i^i  y )  =  ( A  i^i  y ) )
42, 3eqeq12d 2298 . 2  |-  ( x  =  A  ->  ( |^| { x ,  y }  =  ( x  i^i  y )  <->  |^| { A ,  y }  =  ( A  i^i  y
) ) )
5 preq2 3708 . . . 4  |-  ( y  =  B  ->  { A ,  y }  =  { A ,  B }
)
65inteqd 3868 . . 3  |-  ( y  =  B  ->  |^| { A ,  y }  =  |^| { A ,  B } )
7 ineq2 3365 . . 3  |-  ( y  =  B  ->  ( A  i^i  y )  =  ( A  i^i  B
) )
86, 7eqeq12d 2298 . 2  |-  ( y  =  B  ->  ( |^| { A ,  y }  =  ( A  i^i  y )  <->  |^| { A ,  B }  =  ( A  i^i  B ) ) )
9 vex 2792 . . 3  |-  x  e. 
_V
10 vex 2792 . . 3  |-  y  e. 
_V
119, 10intpr 3896 . 2  |-  |^| { x ,  y }  =  ( x  i^i  y
)
124, 8, 11vtocl2g 2848 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  |^| { A ,  B }  =  ( A  i^i  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1685    i^i cin 3152   {cpr 3642   |^|cint 3863
This theorem is referenced by:  intsng  3898  mreincl  13497  subrgin  15564  lssincl  15718  incld  16776  inttop4  24928  inidl  26066
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ral 2549  df-v 2791  df-un 3158  df-in 3160  df-sn 3647  df-pr 3648  df-int 3864
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