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Theorem intpr 3895
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 1580 . . . 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 2791 . . . . . . . 8  |-  y  e. 
_V
32elpr 3658 . . . . . . 7  |-  ( y  e.  { A ,  B }  <->  ( y  =  A  \/  y  =  B ) )
43imbi1i 315 . . . . . 6  |-  ( ( y  e.  { A ,  B }  ->  x  e.  y )  <->  ( (
y  =  A  \/  y  =  B )  ->  x  e.  y ) )
5 jaob 758 . . . . . 6  |-  ( ( ( y  =  A  \/  y  =  B )  ->  x  e.  y )  <->  ( (
y  =  A  ->  x  e.  y )  /\  ( y  =  B  ->  x  e.  y ) ) )
64, 5bitri 240 . . . . 5  |-  ( ( y  e.  { A ,  B }  ->  x  e.  y )  <->  ( (
y  =  A  ->  x  e.  y )  /\  ( y  =  B  ->  x  e.  y ) ) )
76albii 1553 . . . 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 2907 . . . . 5  |-  ( x  e.  A  <->  A. y
( y  =  A  ->  x  e.  y ) )
10 intpr.2 . . . . . 6  |-  B  e. 
_V
1110clel4 2907 . . . . 5  |-  ( x  e.  B  <->  A. y
( y  =  B  ->  x  e.  y ) )
129, 11anbi12i 678 . . . 4  |-  ( ( x  e.  A  /\  x  e.  B )  <->  ( A. y ( y  =  A  ->  x  e.  y )  /\  A. y ( y  =  B  ->  x  e.  y ) ) )
131, 7, 123bitr4i 268 . . 3  |-  ( A. y ( y  e. 
{ A ,  B }  ->  x  e.  y )  <->  ( x  e.  A  /\  x  e.  B ) )
14 vex 2791 . . . 4  |-  x  e. 
_V
1514elint 3868 . . 3  |-  ( x  e.  |^| { A ,  B }  <->  A. y ( y  e.  { A ,  B }  ->  x  e.  y ) )
16 elin 3358 . . 3  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
1713, 15, 163bitr4i 268 . 2  |-  ( x  e.  |^| { A ,  B }  <->  x  e.  ( A  i^i  B ) )
1817eqriv 2280 1  |-  |^| { A ,  B }  =  ( A  i^i  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358   A.wal 1527    = wceq 1623    e. wcel 1684   _Vcvv 2788    i^i cin 3151   {cpr 3641   |^|cint 3862
This theorem is referenced by:  intprg  3896  uniintsn  3899  op1stb  4569  fiint  7133  shincli  21941  chincli  22039  toplat  25290
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-un 3157  df-in 3159  df-sn 3646  df-pr 3647  df-int 3863
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