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Theorem djudisj 5289
 Description: Disjoint unions with disjoint index sets are disjoint. (Contributed by Stefan O'Rear, 21-Nov-2014.)
Assertion
Ref Expression
djudisj
Distinct variable groups:   ,   ,
Allowed substitution hints:   ()   ()   (,)   (,)

Proof of Theorem djudisj
StepHypRef Expression
1 djussxp 5010 . 2
2 incom 3525 . . 3
3 djussxp 5010 . . . 4
4 incom 3525 . . . . 5
5 xpdisj1 5286 . . . . 5
64, 5syl5eq 2479 . . . 4
7 ssdisj 3669 . . . 4
83, 6, 7sylancr 645 . . 3
92, 8syl5eq 2479 . 2
10 ssdisj 3669 . 2
111, 9, 10sylancr 645 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1652  cvv 2948   cin 3311   wss 3312  c0 3620  csn 3806  ciun 4085   cxp 4868 This theorem is referenced by:  ackbij1lem9  8098 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-iun 4087  df-opab 4259  df-xp 4876  df-rel 4877
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