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Theorem disjiun2 37398
 Description: In a disjoint collection, an indexed union is disjoint from an additional term. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
disjiun2.1 Disj
disjiun2.2
disjiun2.3
disjiun2.4
Assertion
Ref Expression
disjiun2
Distinct variable groups:   ,   ,   ,   ,
Allowed substitution hints:   ()   ()

Proof of Theorem disjiun2
StepHypRef Expression
1 disjiun2.3 . . . 4
2 disjiun2.4 . . . . 5
32iunxsng 4360 . . . 4
41, 3syl 17 . . 3
54ineq2d 3634 . 2
6 disjiun2.1 . . 3 Disj
7 disjiun2.2 . . 3
8 eldifi 3555 . . . 4
9 snssi 4116 . . . 4
101, 8, 93syl 18 . . 3
111eldifbd 3417 . . . 4
12 disjsn 4032 . . . 4
1311, 12sylibr 216 . . 3
14 disjiun 4393 . . 3 Disj
156, 7, 10, 13, 14syl13anc 1270 . 2
165, 15eqtr3d 2487 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wceq 1444   wcel 1887   cdif 3401   cin 3403   wss 3404  c0 3731  csn 3968  ciun 4278  Disj wdisj 4373 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431 This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rmo 2745  df-v 3047  df-sbc 3268  df-dif 3407  df-in 3411  df-ss 3418  df-nul 3732  df-sn 3969  df-iun 4280  df-disj 4374 This theorem is referenced by:  caratheodorylem1  38347
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