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Theorem dmin 5034
 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 1616 . . 3
2 vex 2917 . . . . 5
32eldm2 5025 . . . 4
4 elin 3488 . . . . 5
54exbii 1589 . . . 4
63, 5bitri 241 . . 3
7 elin 3488 . . . 4
82eldm2 5025 . . . . 5
92eldm2 5025 . . . . 5
108, 9anbi12i 679 . . . 4
117, 10bitri 241 . . 3
121, 6, 113imtr4i 258 . 2
1312ssriv 3310 1
 Colors of variables: wff set class Syntax hints:   wa 359  wex 1547   wcel 1721   cin 3277   wss 3278  cop 3775   cdm 4835 This theorem is referenced by:  rnin  5238  psssdm2  14598  hauseqcn  24244 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2383 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2389  df-cleq 2395  df-clel 2398  df-nfc 2527  df-rab 2673  df-v 2916  df-dif 3281  df-un 3283  df-in 3285  df-ss 3292  df-nul 3587  df-if 3698  df-sn 3778  df-pr 3779  df-op 3781  df-br 4171  df-dm 4845
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