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Theorem dmin 4742
 Description: The domain of an intersection belong to the intersection of domains. Theorem 6 of [Suppes] p. 60. (Contributed by NM, 15-Sep-2004.)
Assertion
Ref Expression
dmin

Proof of Theorem dmin
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 19.40 1610 . . 3
2 vex 2684 . . . . 5
32eldm2 4732 . . . 4
4 elin 3254 . . . . 5
54exbii 1584 . . . 4
63, 5bitri 183 . . 3
7 elin 3254 . . . 4
82eldm2 4732 . . . . 5
92eldm2 4732 . . . . 5
108, 9anbi12i 455 . . . 4
117, 10bitri 183 . . 3
121, 6, 113imtr4i 200 . 2
1312ssriv 3096 1
 Colors of variables: wff set class Syntax hints:   wa 103  wex 1468   wcel 1480   cin 3065   wss 3066  cop 3525   cdm 4534 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-dm 4544 This theorem is referenced by:  rnin  4943
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