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Theorem intpr 4027
Description: The intersection of a pair is the intersection of its members. Theorem 71 of [Suppes] p. 42. (Contributed by NM, 14-Oct-1999.)
Hypotheses
Ref Expression
intpr.1  |-  A  e. 
_V
intpr.2  |-  B  e. 
_V
Assertion
Ref Expression
intpr  |-  |^| { A ,  B }  =  ( A  i^i  B )

Proof of Theorem intpr
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 19.26 1600 . . . 4  |-  ( A. y ( ( y  =  A  ->  x  e.  y )  /\  (
y  =  B  ->  x  e.  y )
)  <->  ( A. y
( y  =  A  ->  x  e.  y )  /\  A. y
( y  =  B  ->  x  e.  y ) ) )
2 vex 2904 . . . . . . . 8  |-  y  e. 
_V
32elpr 3777 . . . . . . 7  |-  ( y  e.  { A ,  B }  <->  ( y  =  A  \/  y  =  B ) )
43imbi1i 316 . . . . . 6  |-  ( ( y  e.  { A ,  B }  ->  x  e.  y )  <->  ( (
y  =  A  \/  y  =  B )  ->  x  e.  y ) )
5 jaob 759 . . . . . 6  |-  ( ( ( y  =  A  \/  y  =  B )  ->  x  e.  y )  <->  ( (
y  =  A  ->  x  e.  y )  /\  ( y  =  B  ->  x  e.  y ) ) )
64, 5bitri 241 . . . . 5  |-  ( ( y  e.  { A ,  B }  ->  x  e.  y )  <->  ( (
y  =  A  ->  x  e.  y )  /\  ( y  =  B  ->  x  e.  y ) ) )
76albii 1572 . . . 4  |-  ( A. y ( y  e. 
{ A ,  B }  ->  x  e.  y )  <->  A. y ( ( y  =  A  ->  x  e.  y )  /\  ( y  =  B  ->  x  e.  y ) ) )
8 intpr.1 . . . . . 6  |-  A  e. 
_V
98clel4 3020 . . . . 5  |-  ( x  e.  A  <->  A. y
( y  =  A  ->  x  e.  y ) )
10 intpr.2 . . . . . 6  |-  B  e. 
_V
1110clel4 3020 . . . . 5  |-  ( x  e.  B  <->  A. y
( y  =  B  ->  x  e.  y ) )
129, 11anbi12i 679 . . . 4  |-  ( ( x  e.  A  /\  x  e.  B )  <->  ( A. y ( y  =  A  ->  x  e.  y )  /\  A. y ( y  =  B  ->  x  e.  y ) ) )
131, 7, 123bitr4i 269 . . 3  |-  ( A. y ( y  e. 
{ A ,  B }  ->  x  e.  y )  <->  ( x  e.  A  /\  x  e.  B ) )
14 vex 2904 . . . 4  |-  x  e. 
_V
1514elint 4000 . . 3  |-  ( x  e.  |^| { A ,  B }  <->  A. y ( y  e.  { A ,  B }  ->  x  e.  y ) )
16 elin 3475 . . 3  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
1713, 15, 163bitr4i 269 . 2  |-  ( x  e.  |^| { A ,  B }  <->  x  e.  ( A  i^i  B ) )
1817eqriv 2386 1  |-  |^| { A ,  B }  =  ( A  i^i  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359   A.wal 1546    = wceq 1649    e. wcel 1717   _Vcvv 2901    i^i cin 3264   {cpr 3760   |^|cint 3994
This theorem is referenced by:  intprg  4028  uniintsn  4031  op1stb  4700  fiint  7321  shincli  22714  chincli  22812
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 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
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 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-v 2903  df-un 3270  df-in 3272  df-sn 3765  df-pr 3766  df-int 3995
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