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Theorem intpr 4320
 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 1680 . . . 4
2 vex 3112 . . . . . . . 8
32elpr 4047 . . . . . . 7
43imbi1i 325 . . . . . 6
5 jaob 783 . . . . . 6
64, 5bitri 249 . . . . 5
76albii 1640 . . . 4
8 intpr.1 . . . . . 6
98clel4 3239 . . . . 5
10 intpr.2 . . . . . 6
1110clel4 3239 . . . . 5
129, 11anbi12i 697 . . . 4
131, 7, 123bitr4i 277 . . 3
14 vex 3112 . . . 4
1514elint 4292 . . 3
16 elin 3686 . . 3
1713, 15, 163bitr4i 277 . 2
1817eqriv 2453 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  \/wo 368  /\wa 369  A.wal 1393  =wceq 1395  e.wcel 1818   cvv 3109  i^icin 3474  {cpr 4031  |^|cint 4286 This theorem is referenced by:  intprg  4321  uniintsn  4324  op1stb  4722  fiint  7817  shincli  26280  chincli  26378 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-un 3480  df-in 3482  df-sn 4030  df-pr 4032  df-int 4287
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