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Theorem intprg 3639
Description: The intersection of a pair is the intersection of its members. Closed form of intpr 3638. Theorem 71 of [Suppes] p. 42. (Contributed by FL, 27-Apr-2008.)
Assertion
Ref Expression
intprg  V  W  |^| { ,  }  i^i

Proof of Theorem intprg
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 preq1 3438 . . . 4  { ,  }  { ,  }
21inteqd 3611 . . 3  |^| { ,  }  |^| { ,  }
3 ineq1 3125 . . 3  i^i  i^i
42, 3eqeq12d 2051 . 2  |^| { ,  }  i^i  |^| { ,  }  i^i
5 preq2 3439 . . . 4  { ,  }  { ,  }
65inteqd 3611 . . 3  |^| { ,  }  |^| { ,  }
7 ineq2 3126 . . 3  i^i  i^i
86, 7eqeq12d 2051 . 2  |^| { ,  }  i^i  |^| { ,  }  i^i
9 vex 2554 . . 3 
_V
10 vex 2554 . . 3 
_V
119, 10intpr 3638 . 2  |^| { ,  }  i^i
124, 8, 11vtocl2g 2611 1  V  W  |^| { ,  }  i^i
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wceq 1242   wcel 1390    i^i cin 2910   {cpr 3368   |^|cint 3606
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-un 2916  df-in 2918  df-sn 3373  df-pr 3374  df-int 3607
This theorem is referenced by:  intsng  3640  op1stbg  4176
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