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Theorem disjss1 3944
 Description: A subset of a disjoint collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Assertion
Ref Expression
disjss1 Disj Disj
Distinct variable groups:   ,   ,
Allowed substitution hint:   ()

Proof of Theorem disjss1
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 ssel 3118 . . . . . 6
21anim1d 334 . . . . 5
32alrimiv 1851 . . . 4
4 moim 2067 . . . 4
53, 4syl 14 . . 3
65alimdv 1856 . 2
7 dfdisj2 3940 . 2 Disj
8 dfdisj2 3940 . 2 Disj
96, 7, 83imtr4g 204 1 Disj Disj
 Colors of variables: wff set class Syntax hints:   wi 4   wa 103  wal 1330  wmo 2004   wcel 2125   wss 3098  Disj wdisj 3938 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136 This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-rmo 2440  df-in 3104  df-ss 3111  df-disj 3939 This theorem is referenced by:  disjeq1  3945  disjx0  3960  fsumiun  11351
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