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Theorem intpr 3676
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 1411 . . . 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 2605 . . . . . . . 8  |-  y  e. 
_V
32elpr 3427 . . . . . . 7  |-  ( y  e.  { A ,  B }  <->  ( y  =  A  \/  y  =  B ) )
43imbi1i 236 . . . . . 6  |-  ( ( y  e.  { A ,  B }  ->  x  e.  y )  <->  ( (
y  =  A  \/  y  =  B )  ->  x  e.  y ) )
5 jaob 664 . . . . . 6  |-  ( ( ( y  =  A  \/  y  =  B )  ->  x  e.  y )  <->  ( (
y  =  A  ->  x  e.  y )  /\  ( y  =  B  ->  x  e.  y ) ) )
64, 5bitri 182 . . . . 5  |-  ( ( y  e.  { A ,  B }  ->  x  e.  y )  <->  ( (
y  =  A  ->  x  e.  y )  /\  ( y  =  B  ->  x  e.  y ) ) )
76albii 1400 . . . 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 2732 . . . . 5  |-  ( x  e.  A  <->  A. y
( y  =  A  ->  x  e.  y ) )
10 intpr.2 . . . . . 6  |-  B  e. 
_V
1110clel4 2732 . . . . 5  |-  ( x  e.  B  <->  A. y
( y  =  B  ->  x  e.  y ) )
129, 11anbi12i 448 . . . 4  |-  ( ( x  e.  A  /\  x  e.  B )  <->  ( A. y ( y  =  A  ->  x  e.  y )  /\  A. y ( y  =  B  ->  x  e.  y ) ) )
131, 7, 123bitr4i 210 . . 3  |-  ( A. y ( y  e. 
{ A ,  B }  ->  x  e.  y )  <->  ( x  e.  A  /\  x  e.  B ) )
14 vex 2605 . . . 4  |-  x  e. 
_V
1514elint 3650 . . 3  |-  ( x  e.  |^| { A ,  B }  <->  A. y ( y  e.  { A ,  B }  ->  x  e.  y ) )
16 elin 3156 . . 3  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
1713, 15, 163bitr4i 210 . 2  |-  ( x  e.  |^| { A ,  B }  <->  x  e.  ( A  i^i  B ) )
1817eqriv 2079 1  |-  |^| { A ,  B }  =  ( A  i^i  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    \/ wo 662   A.wal 1283    = wceq 1285    e. wcel 1434   _Vcvv 2602    i^i cin 2973   {cpr 3407   |^|cint 3644
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-in 2980  df-sn 3412  df-pr 3413  df-int 3645
This theorem is referenced by:  intprg  3677  op1stb  4235  onintexmid  4323
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