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Theorem disjeq1 4394
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 3497 . . 3  |-  ( A  =  B  ->  B  C_  A )
2 disjss1 4393 . . 3  |-  ( B 
C_  A  ->  (Disj  x  e.  A  C  -> Disj  x  e.  B  C ) )
31, 2syl 17 . 2  |-  ( A  =  B  ->  (Disj  x  e.  A  C  -> Disj  x  e.  B  C ) )
4 eqimss 3496 . . 3  |-  ( A  =  B  ->  A  C_  B )
5 disjss1 4393 . . 3  |-  ( A 
C_  B  ->  (Disj  x  e.  B  C  -> Disj  x  e.  A  C ) )
64, 5syl 17 . 2  |-  ( A  =  B  ->  (Disj  x  e.  B  C  -> Disj  x  e.  A  C ) )
73, 6impbid 195 1  |-  ( A  =  B  ->  (Disj  x  e.  A  C  <-> Disj  x  e.  B  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    = wceq 1455    C_ wss 3416  Disj wdisj 4387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442
This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-rmo 2757  df-in 3423  df-ss 3430  df-disj 4388
This theorem is referenced by:  disjeq1d  4395  volfiniun  22549  disjrnmpt  28244  iundisj2cnt  28424  unelldsys  29029  sigapildsys  29033  ldgenpisyslem1  29034  rossros  29051  measvun  29080  pmeasmono  29206  pmeasadd  29207  meadjuni  38333
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