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Theorem disjss2 4425
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

Proof of Theorem disjss2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 ssel 3497 . . . . 5
21ralimi 2850 . . . 4
3 rmoim 3299 . . . 4
42, 3syl 16 . . 3
54alimdv 1709 . 2
6 df-disj 4423 . 2
7 df-disj 4423 . 2
85, 6, 73imtr4g 270 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  A.wal 1393  e.wcel 1818  A.wral 2807  E*wrmo 2810  C_wss 3475  Disj_wdisj 4422
This theorem is referenced by:  disjeq2  4426  0disj  4445  uniioombllem2  21992  uniioombllem4  21995  disjxwwlks  24736  disjxwwlkn  24745  usgreghash2spotv  25066
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-ral 2812  df-rmo 2815  df-in 3482  df-ss 3489  df-disj 4423
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