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Theorem disji2 4439
Description: Property of a disjoint collection: if ( )= and ( )= , and , then and are disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Hypotheses
Ref Expression
disji.1
disji.2
Assertion
Ref Expression
disji2
Distinct variable groups:   ,   ,   ,   ,   ,

Proof of Theorem disji2
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ne 2654 . . 3
2 disjors 4438 . . . . . 6
3 eqeq1 2461 . . . . . . . 8
4 nfcv 2619 . . . . . . . . . . 11
5 nfcv 2619 . . . . . . . . . . 11
6 disji.1 . . . . . . . . . . 11
74, 5, 6csbhypf 3453 . . . . . . . . . 10
87ineq1d 3698 . . . . . . . . 9
98eqeq1d 2459 . . . . . . . 8
103, 9orbi12d 709 . . . . . . 7
11 eqeq2 2472 . . . . . . . 8
12 nfcv 2619 . . . . . . . . . . 11
13 nfcv 2619 . . . . . . . . . . 11
14 disji.2 . . . . . . . . . . 11
1512, 13, 14csbhypf 3453 . . . . . . . . . 10
1615ineq2d 3699 . . . . . . . . 9
1716eqeq1d 2459 . . . . . . . 8
1811, 17orbi12d 709 . . . . . . 7
1910, 18rspc2v 3219 . . . . . 6
202, 19syl5bi 217 . . . . 5
2120impcom 430 . . . 4
2221ord 377 . . 3
231, 22syl5bi 217 . 2
24233impia 1193 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  ->wi 4  \/wo 368  /\wa 369  /\w3a 973  =wceq 1395  e.wcel 1818  =/=wne 2652  A.wral 2807  [_csb 3434  i^icin 3474   c0 3784  Disj_wdisj 4422
This theorem is referenced by:  disji  4440  disjxiun  4449  voliunlem1  21960
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-v 3111  df-sbc 3328  df-csb 3435  df-dif 3478  df-in 3482  df-nul 3785  df-disj 4423
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