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Theorem dmin 4543
 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 1522 . . 3
2 vex 2560 . . . . 5
32eldm2 4533 . . . 4
4 elin 3126 . . . . 5
54exbii 1496 . . . 4
63, 5bitri 173 . . 3
7 elin 3126 . . . 4
82eldm2 4533 . . . . 5
92eldm2 4533 . . . . 5
108, 9anbi12i 433 . . . 4
117, 10bitri 173 . . 3
121, 6, 113imtr4i 190 . 2
1312ssriv 2949 1
 Colors of variables: wff set class Syntax hints:   wa 97  wex 1381   wcel 1393   cin 2916   wss 2917  cop 3378   cdm 4345 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-dm 4355 This theorem is referenced by:  rnin  4733
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