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Theorem intpr 4292
 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 1727 . . . 4
2 vex 3090 . . . . . . . 8
32elpr 4020 . . . . . . 7
43imbi1i 326 . . . . . 6
5 jaob 790 . . . . . 6
64, 5bitri 252 . . . . 5
76albii 1687 . . . 4
8 intpr.1 . . . . . 6
98clel4 3217 . . . . 5
10 intpr.2 . . . . . 6
1110clel4 3217 . . . . 5
129, 11anbi12i 701 . . . 4
131, 7, 123bitr4i 280 . . 3
14 vex 3090 . . . 4
1514elint 4264 . . 3
16 elin 3655 . . 3
1713, 15, 163bitr4i 280 . 2
1817eqriv 2425 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wo 369   wa 370  wal 1435   wceq 1437   wcel 1870  cvv 3087   cin 3441  cpr 4004  cint 4258 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-v 3089  df-un 3447  df-in 3449  df-sn 4003  df-pr 4005  df-int 4259 This theorem is referenced by:  intprg  4293  uniintsn  4296  op1stb  4692  fiint  7854  shincli  26858  chincli  26956
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