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Theorem intpr 3793
 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
StepHypRef Expression
1 19.26 1592 . . . 4
2 vex 2730 . . . . . . . 8
32elpr 3562 . . . . . . 7
43imbi1i 317 . . . . . 6
5 jaob 761 . . . . . 6
64, 5bitri 242 . . . . 5
76albii 1554 . . . 4
8 intpr.1 . . . . . 6
98clel4 2844 . . . . 5
10 intpr.2 . . . . . 6
1110clel4 2844 . . . . 5
129, 11anbi12i 681 . . . 4
131, 7, 123bitr4i 270 . . 3
14 vex 2730 . . . 4
1514elint 3766 . . 3
16 elin 3266 . . 3
1713, 15, 163bitr4i 270 . 2
1817eqriv 2250 1
 Colors of variables: wff set class Syntax hints:   wi 6   wo 359   wa 360  wal 1532   wceq 1619   wcel 1621  cvv 2727   cin 3077  cpr 3545  cint 3760 This theorem is referenced by:  intprg  3794  uniintsn  3797  op1stb  4460  fiint  7018  shincli  21771  chincli  21869  toplat  24456 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-v 2729  df-un 3083  df-in 3085  df-sn 3550  df-pr 3551  df-int 3761
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