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Theorem disjeq1 4192
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 3403 . . 3  |-  ( A  =  B  ->  B  C_  A )
2 disjss1 4191 . . 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 3402 . . 3  |-  ( A  =  B  ->  A  C_  B )
5 disjss1 4191 . . 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 185 1  |-  ( A  =  B  ->  (Disj  x  e.  A C  <-> Disj  x  e.  B C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    = wceq 1653    C_ wss 3322  Disj wdisj 4185
This theorem is referenced by:  disjeq1d  4193  volfiniun  19446  iundisj2cnt  24160  measvun  24568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-rmo 2715  df-in 3329  df-ss 3336  df-disj 4186
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