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Theorem intprg 3677
 Description: The intersection of a pair is the intersection of its members. Closed form of intpr 3676. Theorem 71 of [Suppes] p. 42. (Contributed by FL, 27-Apr-2008.)
Assertion
Ref Expression
intprg

Proof of Theorem intprg
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 preq1 3477 . . . 4
21inteqd 3649 . . 3
3 ineq1 3167 . . 3
42, 3eqeq12d 2096 . 2
5 preq2 3478 . . . 4
65inteqd 3649 . . 3
7 ineq2 3168 . . 3
86, 7eqeq12d 2096 . 2
9 vex 2605 . . 3
10 vex 2605 . . 3
119, 10intpr 3676 . 2
124, 8, 11vtocl2g 2663 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 102   wceq 1285   wcel 1434   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-ral 2354  df-v 2604  df-un 2978  df-in 2980  df-sn 3412  df-pr 3413  df-int 3645 This theorem is referenced by:  intsng  3678  op1stbg  4236
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