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Theorem dmin 5215
 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 1656 . . 3
2 vex 3121 . . . . 5
32eldm2 5206 . . . 4
4 elin 3692 . . . . 5
54exbii 1644 . . . 4
63, 5bitri 249 . . 3
7 elin 3692 . . . 4
82eldm2 5206 . . . . 5
92eldm2 5206 . . . . 5
108, 9anbi12i 697 . . . 4
117, 10bitri 249 . . 3
121, 6, 113imtr4i 266 . 2
1312ssriv 3513 1
 Colors of variables: wff setvar class Syntax hints:   wa 369  wex 1596   wcel 1767   cin 3480   wss 3481  cop 4038   cdm 5004 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4453  df-dm 5014 This theorem is referenced by:  rnin  5420  psssdm2  15714  hauseqcn  27670
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