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Theorem disjss1 4428
Description: A subset of a disjoint collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Assertion
Ref Expression
disjss1
Distinct variable groups:   ,   ,

Proof of Theorem disjss1
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 ssel 3497 . . . . . 6
21anim1d 564 . . . . 5
32alrimiv 1719 . . . 4
4 moim 2339 . . . 4
53, 4syl 16 . . 3
65alimdv 1709 . 2
7 dfdisj2 4424 . 2
8 dfdisj2 4424 . 2
96, 7, 83imtr4g 270 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  /\wa 369  A.wal 1393  e.wcel 1818  E*wmo 2283  C_wss 3475  Disj_wdisj 4422
This theorem is referenced by:  disjeq1  4429  disjx0  4447  disjxiun  4449  disjss3  4451  volfiniun  21957  uniioovol  21988  uniioombllem4  21995  sibfof  28282
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-rmo 2815  df-in 3482  df-ss 3489  df-disj 4423
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