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Theorem disji 4421
 Description: Property of a disjoint collection: if and have a common element , then . (Contributed by Mario Carneiro, 14-Nov-2016.)
Hypotheses
Ref Expression
disji.1
disji.2
Assertion
Ref Expression
disji Disj
Distinct variable groups:   ,   ,   ,   ,   ,
Allowed substitution hints:   ()   ()

Proof of Theorem disji
StepHypRef Expression
1 inelcm 3863 . 2
2 disji.1 . . . . . 6
3 disji.2 . . . . . 6
42, 3disji2 4420 . . . . 5 Disj
543expia 1197 . . . 4 Disj
65necon1d 2666 . . 3 Disj
763impia 1192 . 2 Disj
81, 7syl3an3 1262 1 Disj
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   w3a 972   wceq 1381   wcel 1802   wne 2636   cin 3457  c0 3767  Disj wdisj 4403 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-in 3465  df-nul 3768  df-disj 4404 This theorem is referenced by:  volfiniun  21823
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