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Theorem intprg 4048
Description: The intersection of a pair is the intersection of its members. Closed form of intpr 4047. 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 3847 . . . 4  |-  ( x  =  A  ->  { x ,  y }  =  { A ,  y } )
21inteqd 4019 . . 3  |-  ( x  =  A  ->  |^| { x ,  y }  =  |^| { A ,  y } )
3 ineq1 3499 . . 3  |-  ( x  =  A  ->  (
x  i^i  y )  =  ( A  i^i  y ) )
42, 3eqeq12d 2422 . 2  |-  ( x  =  A  ->  ( |^| { x ,  y }  =  ( x  i^i  y )  <->  |^| { A ,  y }  =  ( A  i^i  y
) ) )
5 preq2 3848 . . . 4  |-  ( y  =  B  ->  { A ,  y }  =  { A ,  B }
)
65inteqd 4019 . . 3  |-  ( y  =  B  ->  |^| { A ,  y }  =  |^| { A ,  B } )
7 ineq2 3500 . . 3  |-  ( y  =  B  ->  ( A  i^i  y )  =  ( A  i^i  B
) )
86, 7eqeq12d 2422 . 2  |-  ( y  =  B  ->  ( |^| { A ,  y }  =  ( A  i^i  y )  <->  |^| { A ,  B }  =  ( A  i^i  B ) ) )
9 vex 2923 . . 3  |-  x  e. 
_V
10 vex 2923 . . 3  |-  y  e. 
_V
119, 10intpr 4047 . 2  |-  |^| { x ,  y }  =  ( x  i^i  y
)
124, 8, 11vtocl2g 2979 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  |^| { A ,  B }  =  ( A  i^i  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721    i^i cin 3283   {cpr 3779   |^|cint 4014
This theorem is referenced by:  intsng  4049  inelfi  7385  mreincl  13783  subrgin  15850  lssincl  16000  incld  17066  difelsiga  24473  inidl  26534
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ral 2675  df-v 2922  df-un 3289  df-in 3291  df-sn 3784  df-pr 3785  df-int 4015
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