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Theorem disji2 3958
 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 disjnims 3957 . . 3 Disj
2 neeq1 2340 . . . . 5
3 nfcv 2299 . . . . . . . 8
4 nfcv 2299 . . . . . . . 8
5 disji.1 . . . . . . . 8
63, 4, 5csbhypf 3069 . . . . . . 7
76ineq1d 3307 . . . . . 6
87eqeq1d 2166 . . . . 5
92, 8imbi12d 233 . . . 4
10 neeq2 2341 . . . . 5
11 nfcv 2299 . . . . . . . 8
12 nfcv 2299 . . . . . . . 8
13 disji.2 . . . . . . . 8
1411, 12, 13csbhypf 3069 . . . . . . 7
1514ineq2d 3308 . . . . . 6
1615eqeq1d 2166 . . . . 5
1710, 16imbi12d 233 . . . 4
189, 17rspc2v 2829 . . 3
191, 18mpan9 279 . 2 Disj
20193impia 1182 1 Disj
 Colors of variables: wff set class Syntax hints:   wi 4   wa 103   w3a 963   wceq 1335   wcel 2128   wne 2327  wral 2435  csb 3031   cin 3101  c0 3394  Disj wdisj 3942 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-in 3108  df-nul 3395  df-disj 3943 This theorem is referenced by: (None)
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