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Theorem intpr 4075
 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 1603 . . . 4
2 vex 2951 . . . . . . . 8
32elpr 3824 . . . . . . 7
43imbi1i 316 . . . . . 6
5 jaob 759 . . . . . 6
64, 5bitri 241 . . . . 5
76albii 1575 . . . 4
8 intpr.1 . . . . . 6
98clel4 3067 . . . . 5
10 intpr.2 . . . . . 6
1110clel4 3067 . . . . 5
129, 11anbi12i 679 . . . 4
131, 7, 123bitr4i 269 . . 3
14 vex 2951 . . . 4
1514elint 4048 . . 3
16 elin 3522 . . 3
1713, 15, 163bitr4i 269 . 2
1817eqriv 2432 1
 Colors of variables: wff set class Syntax hints:   wi 4   wo 358   wa 359  wal 1549   wceq 1652   wcel 1725  cvv 2948   cin 3311  cpr 3807  cint 4042 This theorem is referenced by:  intprg  4076  uniintsn  4079  op1stb  4750  fiint  7375  shincli  22856  chincli  22954 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-un 3317  df-in 3319  df-sn 3812  df-pr 3813  df-int 4043
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