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Theorem disjeq1 4157
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 3369 . . 3  |-  ( A  =  B  ->  B  C_  A )
2 disjss1 4156 . . 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 3368 . . 3  |-  ( A  =  B  ->  A  C_  B )
5 disjss1 4156 . . 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 184 1  |-  ( A  =  B  ->  (Disj  x  e.  A C  <-> Disj  x  e.  B C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649    C_ wss 3288  Disj wdisj 4150
This theorem is referenced by:  disjeq1d  4158  volfiniun  19402  iundisj2cnt  24116  measvun  24524
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-rmo 2682  df-in 3295  df-ss 3302  df-disj 4151
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