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Theorem disjss1 4190
 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 3344 . . . . . 6
21anim1d 549 . . . . 5
32alrimiv 1642 . . . 4
4 moim 2329 . . . 4
53, 4syl 16 . . 3
65alimdv 1632 . 2
7 dfdisj2 4186 . 2 Disj
8 dfdisj2 4186 . 2 Disj
96, 7, 83imtr4g 263 1 Disj Disj
 Colors of variables: wff set class Syntax hints:   wi 4   wa 360  wal 1550   wcel 1726  wmo 2284   wss 3322  Disj wdisj 4184 This theorem is referenced by:  disjeq1  4191  disjx0  4209  disjxiun  4211  disjss3  4213  volfiniun  19443  uniioovol  19473  uniioombllem4  19480  sibfof  24656 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-rmo 2715  df-in 3329  df-ss 3336  df-disj 4185
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