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Theorem disjnim 3929
 Description: If a collection for is disjoint, then pairs are disjoint. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by Jim Kingdon, 6-Oct-2022.)
Hypothesis
Ref Expression
disjnim.1
Assertion
Ref Expression
disjnim Disj
Distinct variable groups:   ,,   ,   ,
Allowed substitution hints:   ()   ()

Proof of Theorem disjnim
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 df-disj 3916 . 2 Disj
2 disjnim.1 . . . . . . 7
32eleq2d 2210 . . . . . 6
43rmo4 2882 . . . . 5
54albii 1447 . . . 4
6 ralcom4 2712 . . . 4
75, 6bitr4i 186 . . 3
8 ralcom4 2712 . . . . 5
9 19.23v 1856 . . . . . . . . 9
109biimpi 119 . . . . . . . 8
1110necon3ad 2351 . . . . . . 7
12 notm0 3389 . . . . . . . 8
13 elin 3265 . . . . . . . . . 10
1413exbii 1585 . . . . . . . . 9
1514notbii 658 . . . . . . . 8
1612, 15bitr3i 185 . . . . . . 7
1711, 16syl6ibr 161 . . . . . 6
1817ralimi 2499 . . . . 5
198, 18sylbir 134 . . . 4
2019ralimi 2499 . . 3
217, 20sylbi 120 . 2
221, 21sylbi 120 1 Disj
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 103  wal 1330   wceq 1332  wex 1469   wcel 1481   wne 2309  wral 2417  wrmo 2420   cin 3076  c0 3369  Disj wdisj 3915 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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rmo 2425  df-v 2692  df-dif 3079  df-in 3083  df-nul 3370  df-disj 3916 This theorem is referenced by:  disjnims  3930
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