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Theorem disjin 23980
 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 Disj Disj

Proof of Theorem disjin
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 inss1 3521 . . . . . . 7
21sseli 3304 . . . . . 6
32anim2i 553 . . . . 5
43ax-gen 1552 . . . 4
54rmoimi2 3095 . . 3
65alimi 1565 . 2
7 df-disj 4143 . 2 Disj
8 df-disj 4143 . 2 Disj
96, 7, 83imtr4i 258 1 Disj Disj
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359  wal 1546   wcel 1721  wrmo 2669   cin 3279  Disj wdisj 4142 This theorem is referenced by:  measinblem  24527 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 2385 This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-rmo 2674  df-v 2918  df-in 3287  df-ss 3294  df-disj 4143
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