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Theorem intprg 4076
Description: The intersection of a pair is the intersection of its members. Closed form of intpr 4075. 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 3875 . . . 4  |-  ( x  =  A  ->  { x ,  y }  =  { A ,  y } )
21inteqd 4047 . . 3  |-  ( x  =  A  ->  |^| { x ,  y }  =  |^| { A ,  y } )
3 ineq1 3527 . . 3  |-  ( x  =  A  ->  (
x  i^i  y )  =  ( A  i^i  y ) )
42, 3eqeq12d 2449 . 2  |-  ( x  =  A  ->  ( |^| { x ,  y }  =  ( x  i^i  y )  <->  |^| { A ,  y }  =  ( A  i^i  y
) ) )
5 preq2 3876 . . . 4  |-  ( y  =  B  ->  { A ,  y }  =  { A ,  B }
)
65inteqd 4047 . . 3  |-  ( y  =  B  ->  |^| { A ,  y }  =  |^| { A ,  B } )
7 ineq2 3528 . . 3  |-  ( y  =  B  ->  ( A  i^i  y )  =  ( A  i^i  B
) )
86, 7eqeq12d 2449 . 2  |-  ( y  =  B  ->  ( |^| { A ,  y }  =  ( A  i^i  y )  <->  |^| { A ,  B }  =  ( A  i^i  B ) ) )
9 vex 2951 . . 3  |-  x  e. 
_V
10 vex 2951 . . 3  |-  y  e. 
_V
119, 10intpr 4075 . 2  |-  |^| { x ,  y }  =  ( x  i^i  y
)
124, 8, 11vtocl2g 3007 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 1652    e. wcel 1725    i^i cin 3311   {cpr 3807   |^|cint 4042
This theorem is referenced by:  intsng  4077  inelfi  7415  mreincl  13816  subrgin  15883  lssincl  16033  incld  17099  difelsiga  24508  inidl  26631
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-v 2950  df-un 3317  df-in 3319  df-sn 3812  df-pr 3813  df-int 4043
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