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Theorem disjeq1 4414
Description: Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Assertion
Ref Expression
disjeq1  |-  ( A  =  B  ->  (Disj  x  e.  A  C  <-> Disj  x  e.  B  C ) )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    C( x)

Proof of Theorem disjeq1
StepHypRef Expression
1 eqimss2 3542 . . 3  |-  ( A  =  B  ->  B  C_  A )
2 disjss1 4413 . . 3  |-  ( B 
C_  A  ->  (Disj  x  e.  A  C  -> Disj  x  e.  B  C ) )
31, 2syl 16 . 2  |-  ( A  =  B  ->  (Disj  x  e.  A  C  -> Disj  x  e.  B  C ) )
4 eqimss 3541 . . 3  |-  ( A  =  B  ->  A  C_  B )
5 disjss1 4413 . . 3  |-  ( A 
C_  B  ->  (Disj  x  e.  B  C  -> Disj  x  e.  A  C ) )
64, 5syl 16 . 2  |-  ( A  =  B  ->  (Disj  x  e.  B  C  -> Disj  x  e.  A  C ) )
73, 6impbid 191 1  |-  ( A  =  B  ->  (Disj  x  e.  A  C  <-> Disj  x  e.  B  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1383    C_ wss 3461  Disj wdisj 4407
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-rmo 2801  df-in 3468  df-ss 3475  df-disj 4408
This theorem is referenced by:  disjeq1d  4415  volfiniun  21935  iundisj2cnt  27582  measvun  28158
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