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Theorem intpr 3836
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
StepHypRef Expression
1 19.26 1592 . . . 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 2743 . . . . . . . 8  |-  y  e. 
_V
32elpr 3599 . . . . . . 7  |-  ( y  e.  { A ,  B }  <->  ( y  =  A  \/  y  =  B ) )
43imbi1i 317 . . . . . 6  |-  ( ( y  e.  { A ,  B }  ->  x  e.  y )  <->  ( (
y  =  A  \/  y  =  B )  ->  x  e.  y ) )
5 jaob 761 . . . . . 6  |-  ( ( ( y  =  A  \/  y  =  B )  ->  x  e.  y )  <->  ( (
y  =  A  ->  x  e.  y )  /\  ( y  =  B  ->  x  e.  y ) ) )
64, 5bitri 242 . . . . 5  |-  ( ( y  e.  { A ,  B }  ->  x  e.  y )  <->  ( (
y  =  A  ->  x  e.  y )  /\  ( y  =  B  ->  x  e.  y ) ) )
76albii 1554 . . . 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 2858 . . . . 5  |-  ( x  e.  A  <->  A. y
( y  =  A  ->  x  e.  y ) )
10 intpr.2 . . . . . 6  |-  B  e. 
_V
1110clel4 2858 . . . . 5  |-  ( x  e.  B  <->  A. y
( y  =  B  ->  x  e.  y ) )
129, 11anbi12i 681 . . . 4  |-  ( ( x  e.  A  /\  x  e.  B )  <->  ( A. y ( y  =  A  ->  x  e.  y )  /\  A. y ( y  =  B  ->  x  e.  y ) ) )
131, 7, 123bitr4i 270 . . 3  |-  ( A. y ( y  e. 
{ A ,  B }  ->  x  e.  y )  <->  ( x  e.  A  /\  x  e.  B ) )
14 vex 2743 . . . 4  |-  x  e. 
_V
1514elint 3809 . . 3  |-  ( x  e.  |^| { A ,  B }  <->  A. y ( y  e.  { A ,  B }  ->  x  e.  y ) )
16 elin 3300 . . 3  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
1713, 15, 163bitr4i 270 . 2  |-  ( x  e.  |^| { A ,  B }  <->  x  e.  ( A  i^i  B ) )
1817eqriv 2253 1  |-  |^| { A ,  B }  =  ( A  i^i  B )
Colors of variables: wff set class
Syntax hints:    -> wi 6    \/ wo 359    /\ wa 360   A.wal 1532    = wceq 1619    e. wcel 1621   _Vcvv 2740    i^i cin 3093   {cpr 3582   |^|cint 3803
This theorem is referenced by:  intprg  3837  uniintsn  3840  op1stb  4506  fiint  7066  shincli  21866  chincli  21964  toplat  24622
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-v 2742  df-un 3099  df-in 3101  df-sn 3587  df-pr 3588  df-int 3804
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