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Theorem disjeq1 4000
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 3231 . . 3  |-  ( A  =  B  ->  B  C_  A )
2 disjss1 3999 . . 3  |-  ( B 
C_  A  ->  (Disj  x  e.  A C  -> Disj  x  e.  B C ) )
31, 2syl 15 . 2  |-  ( A  =  B  ->  (Disj  x  e.  A C  -> Disj  x  e.  B C ) )
4 eqimss 3230 . . 3  |-  ( A  =  B  ->  A  C_  B )
5 disjss1 3999 . . 3  |-  ( A 
C_  B  ->  (Disj  x  e.  B C  -> Disj  x  e.  A C ) )
64, 5syl 15 . 2  |-  ( A  =  B  ->  (Disj  x  e.  B C  -> Disj  x  e.  A C ) )
73, 6impbid 183 1  |-  ( A  =  B  ->  (Disj  x  e.  A C  <-> Disj  x  e.  B C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    C_ wss 3152  Disj wdisj 3993
This theorem is referenced by:  disjeq1d  4001  volfiniun  18904  iundisj2cnt  23368  measvun  23537
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-rmo 2551  df-in 3159  df-ss 3166  df-disj 3994
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