Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  disji Structured version   Unicode version

Theorem disji 4202
 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 3684 . 2
2 disji.1 . . . . . 6
3 disji.2 . . . . . 6
42, 3disji2 4201 . . . . 5 Disj
543expia 1156 . . . 4 Disj
65necon1d 2675 . . 3 Disj
763impia 1151 . 2 Disj
81, 7syl3an3 1220 1 Disj
 Colors of variables: wff set class Syntax hints:   wi 4   wa 360   w3a 937   wceq 1653   wcel 1726   wne 2601   cin 3321  c0 3630  Disj wdisj 4184 This theorem is referenced by:  volfiniun  19443 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-3an 939  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-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-in 3329  df-nul 3631  df-disj 4185
 Copyright terms: Public domain W3C validator