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Theorem disjin 24072
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 3553 . . . . . . 7
21sseli 3337 . . . . . 6
32anim2i 554 . . . . 5
43ax-gen 1556 . . . 4
54rmoimi2 3148 . . 3
65alimi 1569 . 2
7 df-disj 4218 . 2
8 df-disj 4218 . 2
96, 7, 83imtr4i 259 1
Colors of variables: wff set class
Syntax hints:  ->wi 4  /\wa 360  A.wal 1550  e.wcel 1728  E*wrmo 2719  i^icin 3312  Disj_wdisj 4217
This theorem is referenced by:  measinblem  24804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1955  ax-ext 2428
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2296  df-mo 2297  df-clab 2434  df-cleq 2440  df-clel 2443  df-nfc 2572  df-rmo 2724  df-v 2971  df-in 3320  df-ss 3327  df-disj 4218
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