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Theorem disji2 4429
 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 Disj
Distinct variable groups:   ,   ,   ,   ,   ,
Allowed substitution hint:   ()

Proof of Theorem disji2
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ne 2659 . . 3
2 disjors 4428 . . . . . 6 Disj
3 eqeq1 2466 . . . . . . . 8
4 nfcv 2624 . . . . . . . . . . 11
5 nfcv 2624 . . . . . . . . . . 11
6 disji.1 . . . . . . . . . . 11
74, 5, 6csbhypf 3449 . . . . . . . . . 10
87ineq1d 3694 . . . . . . . . 9
98eqeq1d 2464 . . . . . . . 8
103, 9orbi12d 709 . . . . . . 7
11 eqeq2 2477 . . . . . . . 8
12 nfcv 2624 . . . . . . . . . . 11
13 nfcv 2624 . . . . . . . . . . 11
14 disji.2 . . . . . . . . . . 11
1512, 13, 14csbhypf 3449 . . . . . . . . . 10
1615ineq2d 3695 . . . . . . . . 9
1716eqeq1d 2464 . . . . . . . 8
1811, 17orbi12d 709 . . . . . . 7
1910, 18rspc2v 3218 . . . . . 6
202, 19syl5bi 217 . . . . 5 Disj
2120impcom 430 . . . 4 Disj
2221ord 377 . . 3 Disj
231, 22syl5bi 217 . 2 Disj
24233impia 1188 1 Disj
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wo 368   wa 369   w3a 968   wceq 1374   wcel 1762   wne 2657  wral 2809  csb 3430   cin 3470  c0 3780  Disj wdisj 4412 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-in 3478  df-nul 3781  df-disj 4413 This theorem is referenced by:  disji  4430  disjxiun  4439  voliunlem1  21690
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