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Theorem disjnims 3916
Description: If a collection  B ( i ) for  i  e.  A 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  x  e.  A  B  ->  A. i  e.  A  A. j  e.  A  (
i  =/=  j  -> 
( [_ i  /  x ]_ B  i^i  [_ j  /  x ]_ B )  =  (/) ) )
Distinct variable groups:    i, j, x, A    B, i, j
Allowed substitution hint:    B( x)

Proof of Theorem disjnims
StepHypRef Expression
1 nfcv 2279 . . 3  |-  F/_ i B
2 nfcsb1v 3030 . . 3  |-  F/_ x [_ i  /  x ]_ B
3 csbeq1a 3007 . . 3  |-  ( x  =  i  ->  B  =  [_ i  /  x ]_ B )
41, 2, 3cbvdisj 3911 . 2  |-  (Disj  x  e.  A  B  <-> Disj  i  e.  A  [_ i  /  x ]_ B )
5 csbeq1 3001 . . 3  |-  ( i  =  j  ->  [_ i  /  x ]_ B  = 
[_ j  /  x ]_ B )
65disjnim 3915 . 2  |-  (Disj  i  e.  A  [_ i  /  x ]_ B  ->  A. i  e.  A  A. j  e.  A  ( i  =/=  j  ->  ( [_ i  /  x ]_ B  i^i  [_ j  /  x ]_ B )  =  (/) ) )
74, 6sylbi 120 1  |-  (Disj  x  e.  A  B  ->  A. i  e.  A  A. j  e.  A  (
i  =/=  j  -> 
( [_ i  /  x ]_ B  i^i  [_ j  /  x ]_ B )  =  (/) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331    =/= wne 2306   A.wral 2414   [_csb 2998    i^i 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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