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Theorem disjnims 3916
 Description: If a collection for is disjoint, then pairs are disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.) (Revised by Jim Kingdon, 7-Oct-2022.)
Assertion
Ref Expression
disjnims Disj
Distinct variable groups:   ,,,   ,,
Allowed substitution hint:   ()

Proof of Theorem disjnims
StepHypRef Expression
1 nfcv 2279 . . 3
2 nfcsb1v 3030 . . 3
3 csbeq1a 3007 . . 3
41, 2, 3cbvdisj 3911 . 2 Disj Disj
5 csbeq1 3001 . . 3
65disjnim 3915 . 2 Disj
74, 6sylbi 120 1 Disj
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1331   wne 2306  wral 2414  csb 2998   cin 3065  c0 3358  Disj wdisj 3901 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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rmo 2422  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-in 3072  df-nul 3359  df-disj 3902 This theorem is referenced by:  disji2  3917
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