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Theorem disjss2 3748
 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 2939 . . . . 5
21ralimi 2384 . . . 4
3 rmoim 2740 . . . 4
42, 3syl 14 . . 3
54alimdv 1759 . 2
6 df-disj 3746 . 2 Disj
7 df-disj 3746 . 2 Disj
85, 6, 73imtr4g 194 1 Disj Disj
 Colors of variables: wff set class Syntax hints:   wi 4  wal 1241   wcel 1393  wral 2306  wrmo 2309   wss 2917  Disj wdisj 3745 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-ral 2311  df-rmo 2314  df-in 2924  df-ss 2931  df-disj 3746 This theorem is referenced by:  disjeq2  3749  0disj  3761
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