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Theorem intpr 4075
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 1603 . . . 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 2951 . . . . . . . 8  |-  y  e. 
_V
32elpr 3824 . . . . . . 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 1575 . . . 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 3067 . . . . 5  |-  ( x  e.  A  <->  A. y
( y  =  A  ->  x  e.  y ) )
10 intpr.2 . . . . . 6  |-  B  e. 
_V
1110clel4 3067 . . . . 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 2951 . . . 4  |-  x  e. 
_V
1514elint 4048 . . 3  |-  ( x  e.  |^| { A ,  B }  <->  A. y ( y  e.  { A ,  B }  ->  x  e.  y ) )
16 elin 3522 . . 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 2432 1  |-  |^| { A ,  B }  =  ( A  i^i  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359   A.wal 1549    = wceq 1652    e. wcel 1725   _Vcvv 2948    i^i cin 3311   {cpr 3807   |^|cint 4042
This theorem is referenced by:  intprg  4076  uniintsn  4079  op1stb  4750  fiint  7375  shincli  22856  chincli  22954
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-un 3317  df-in 3319  df-sn 3812  df-pr 3813  df-int 4043
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