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Theorem intprg 4323
Description: The intersection of a pair is the intersection of its members. Closed form of intpr 4322. 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 4111 . . . 4  |-  ( x  =  A  ->  { x ,  y }  =  { A ,  y } )
21inteqd 4293 . . 3  |-  ( x  =  A  ->  |^| { x ,  y }  =  |^| { A ,  y } )
3 ineq1 3689 . . 3  |-  ( x  =  A  ->  (
x  i^i  y )  =  ( A  i^i  y ) )
42, 3eqeq12d 2479 . 2  |-  ( x  =  A  ->  ( |^| { x ,  y }  =  ( x  i^i  y )  <->  |^| { A ,  y }  =  ( A  i^i  y
) ) )
5 preq2 4112 . . . 4  |-  ( y  =  B  ->  { A ,  y }  =  { A ,  B }
)
65inteqd 4293 . . 3  |-  ( y  =  B  ->  |^| { A ,  y }  =  |^| { A ,  B } )
7 ineq2 3690 . . 3  |-  ( y  =  B  ->  ( A  i^i  y )  =  ( A  i^i  B
) )
86, 7eqeq12d 2479 . 2  |-  ( y  =  B  ->  ( |^| { A ,  y }  =  ( A  i^i  y )  <->  |^| { A ,  B }  =  ( A  i^i  B ) ) )
9 vex 3112 . . 3  |-  x  e. 
_V
10 vex 3112 . . 3  |-  y  e. 
_V
119, 10intpr 4322 . 2  |-  |^| { x ,  y }  =  ( x  i^i  y
)
124, 8, 11vtocl2g 3171 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  |^| { A ,  B }  =  ( A  i^i  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819    i^i cin 3470   {cpr 4034   |^|cint 4288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-v 3111  df-un 3476  df-in 3478  df-sn 4033  df-pr 4035  df-int 4289
This theorem is referenced by:  intsng  4324  inelfi  7896  mreincl  15016  subrgin  17579  lssincl  17738  incld  19671  difelsiga  28306  inidl  30632
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