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Theorem intpr 3676
 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 1411 . . . 4
2 vex 2605 . . . . . . . 8
32elpr 3427 . . . . . . 7
43imbi1i 236 . . . . . 6
5 jaob 664 . . . . . 6
64, 5bitri 182 . . . . 5
76albii 1400 . . . 4
8 intpr.1 . . . . . 6
98clel4 2732 . . . . 5
10 intpr.2 . . . . . 6
1110clel4 2732 . . . . 5
129, 11anbi12i 448 . . . 4
131, 7, 123bitr4i 210 . . 3
14 vex 2605 . . . 4
1514elint 3650 . . 3
16 elin 3156 . . 3
1713, 15, 163bitr4i 210 . 2
1817eqriv 2079 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 102   wo 662  wal 1283   wceq 1285   wcel 1434  cvv 2602   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-v 2604  df-un 2978  df-in 2980  df-sn 3412  df-pr 3413  df-int 3645 This theorem is referenced by:  intprg  3677  op1stb  4235  onintexmid  4323
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