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Theorem intprg 3912
Description: The intersection of a pair is the intersection of its members. Closed form of intpr 3911. 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 3719 . . . 4  |-  ( x  =  A  ->  { x ,  y }  =  { A ,  y } )
21inteqd 3883 . . 3  |-  ( x  =  A  ->  |^| { x ,  y }  =  |^| { A ,  y } )
3 ineq1 3376 . . 3  |-  ( x  =  A  ->  (
x  i^i  y )  =  ( A  i^i  y ) )
42, 3eqeq12d 2310 . 2  |-  ( x  =  A  ->  ( |^| { x ,  y }  =  ( x  i^i  y )  <->  |^| { A ,  y }  =  ( A  i^i  y
) ) )
5 preq2 3720 . . . 4  |-  ( y  =  B  ->  { A ,  y }  =  { A ,  B }
)
65inteqd 3883 . . 3  |-  ( y  =  B  ->  |^| { A ,  y }  =  |^| { A ,  B } )
7 ineq2 3377 . . 3  |-  ( y  =  B  ->  ( A  i^i  y )  =  ( A  i^i  B
) )
86, 7eqeq12d 2310 . 2  |-  ( y  =  B  ->  ( |^| { A ,  y }  =  ( A  i^i  y )  <->  |^| { A ,  B }  =  ( A  i^i  B ) ) )
9 vex 2804 . . 3  |-  x  e. 
_V
10 vex 2804 . . 3  |-  y  e. 
_V
119, 10intpr 3911 . 2  |-  |^| { x ,  y }  =  ( x  i^i  y
)
124, 8, 11vtocl2g 2860 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 1632    e. wcel 1696    i^i cin 3164   {cpr 3654   |^|cint 3878
This theorem is referenced by:  intsng  3913  mreincl  13517  subrgin  15584  lssincl  15738  incld  16796  difelsiga  23509  inttop4  25620  inidl  26758
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-v 2803  df-un 3170  df-in 3172  df-sn 3659  df-pr 3660  df-int 3879
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