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Theorem intpr 3895
 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
intpr.2
Assertion
Ref Expression
intpr

Proof of Theorem intpr
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 19.26 1580 . . . 4
2 vex 2791 . . . . . . . 8
32elpr 3658 . . . . . . 7
43imbi1i 315 . . . . . 6
5 jaob 758 . . . . . 6
64, 5bitri 240 . . . . 5
76albii 1553 . . . 4
8 intpr.1 . . . . . 6
98clel4 2907 . . . . 5
10 intpr.2 . . . . . 6
1110clel4 2907 . . . . 5
129, 11anbi12i 678 . . . 4
131, 7, 123bitr4i 268 . . 3
14 vex 2791 . . . 4
1514elint 3868 . . 3
16 elin 3358 . . 3
1713, 15, 163bitr4i 268 . 2
1817eqriv 2280 1
 Colors of variables: wff set class Syntax hints:   wi 4   wo 357   wa 358  wal 1527   wceq 1623   wcel 1684  cvv 2788   cin 3151  cpr 3641  cint 3862 This theorem is referenced by:  intprg  3896  uniintsn  3899  op1stb  4569  fiint  7133  shincli  21941  chincli  22039  toplat  25290 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-un 3157  df-in 3159  df-sn 3646  df-pr 3647  df-int 3863
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