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Theorem disjin 24548
Description: If a collection is disjoint, so is the collection of the intersections with a given set. (Contributed by Thierry Arnoux, 14-Feb-2017.)
Assertion
Ref Expression
disjin

Proof of Theorem disjin
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 inss1 3593 . . . . . . 7
21sseli 3377 . . . . . 6
32anim2i 554 . . . . 5
43ax-gen 1562 . . . 4
54rmoimi2 3186 . . 3
65alimi 1575 . 2
7 df-disj 4273 . 2
8 df-disj 4273 . 2
96, 7, 83imtr4i 259 1
Colors of variables: wff set class
Syntax hints:  ->wi 4  /\wa 360  A.wal 1556  e.wcel 1724  E*wrmo 2754  i^icin 3352  Disj_wdisj 4272
This theorem is referenced by:  measinblem  25305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1562  ax-4 1573  ax-5 1636  ax-6 1677  ax-7 1697  ax-10 1743  ax-11 1748  ax-12 1760  ax-13 1947  ax-ext 2462
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1337  df-ex 1558  df-nf 1561  df-sb 1669  df-eu 2309  df-mo 2310  df-clab 2468  df-cleq 2474  df-clel 2477  df-nfc 2606  df-rmo 2759  df-v 3008  df-in 3360  df-ss 3367  df-disj 4273
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