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Theorem disjeq1 4358
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 3483 . . 3  |-  ( A  =  B  ->  B  C_  A )
2 disjss1 4357 . . 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 3482 . . 3  |-  ( A  =  B  ->  A  C_  B )
5 disjss1 4357 . . 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 1399    C_ wss 3402  Disj wdisj 4351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2016  ax-ext 2370
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2232  df-mo 2233  df-clab 2378  df-cleq 2384  df-clel 2387  df-rmo 2750  df-in 3409  df-ss 3416  df-disj 4352
This theorem is referenced by:  disjeq1d  4359  volfiniun  22061  disjrnmpt  27604  iundisj2cnt  27787  unelldsys  28338  sigapildsys  28342  ldgenpisyslem  28343  measvun  28372
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