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Theorem disjss2 4260
 Description: If each element of a collection is contained in a disjoint collection, the original collection is also disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Assertion
Ref Expression
disjss2 Disj Disj

Proof of Theorem disjss2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 ssel 3345 . . . . 5
21ralimi 2786 . . . 4
3 rmoim 3153 . . . 4
42, 3syl 16 . . 3
54alimdv 1675 . 2
6 df-disj 4258 . 2 Disj
7 df-disj 4258 . 2 Disj
85, 6, 73imtr4g 270 1 Disj Disj
 Colors of variables: wff setvar class Syntax hints:   wi 4  wal 1367   wcel 1756  wral 2710  wrmo 2713   wss 3323  Disj wdisj 4257 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419 This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-ral 2715  df-rmo 2718  df-in 3330  df-ss 3337  df-disj 4258 This theorem is referenced by:  disjeq2  4261  0disj  4280  uniioombllem2  21038  uniioombllem4  21041  disjxwwlks  30321  disjxwwlkn  30517  usgreghash2spotv  30612
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