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Theorem disjss2 4371
 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 3438 . . . . 5
21ralimi 2799 . . . 4
3 rmoim 3251 . . . 4
42, 3syl 17 . . 3
54alimdv 1732 . 2
6 df-disj 4369 . 2 Disj
7 df-disj 4369 . 2 Disj
85, 6, 73imtr4g 272 1 Disj Disj
 Colors of variables: wff setvar class Syntax hints:   wi 4  wal 1405   wcel 1844  wral 2756  wrmo 2759   wss 3416  Disj wdisj 4368 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382 This theorem depends on definitions:  df-bi 187  df-an 371  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-ral 2761  df-rmo 2764  df-in 3423  df-ss 3430  df-disj 4369 This theorem is referenced by:  disjeq2  4372  0disj  4390  uniioombllem2  22286  uniioombllem4  22289  disjxwwlks  25165  disjxwwlkn  25174  usgreghash2spotv  25495
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