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Theorem decidin 11054
Description: If A is a decidable subclass of B (meaning: it is a subclass of B and it is decidable in B), and B is decidable in C, then A is decidable in C. (Contributed by BJ, 19-Feb-2022.)
Hypotheses
Ref Expression
decidin.ss  |-  ( ph  ->  A  C_  B )
decidin.a  |-  ( ph  ->  A DECIDin  B )
decidin.b  |-  ( ph  ->  B DECIDin  C )
Assertion
Ref Expression
decidin  |-  ( ph  ->  A DECIDin  C )

Proof of Theorem decidin
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 decidin.b . . . 4  |-  ( ph  ->  B DECIDin  C )
2 decidi 11052 . . . 4  |-  ( B DECIDin  C  -> 
( x  e.  C  ->  ( x  e.  B  \/  -.  x  e.  B
) ) )
31, 2syl 14 . . 3  |-  ( ph  ->  ( x  e.  C  ->  ( x  e.  B  \/  -.  x  e.  B
) ) )
4 decidin.a . . . . 5  |-  ( ph  ->  A DECIDin  B )
5 decidi 11052 . . . . 5  |-  ( A DECIDin  B  -> 
( x  e.  B  ->  ( x  e.  A  \/  -.  x  e.  A
) ) )
64, 5syl 14 . . . 4  |-  ( ph  ->  ( x  e.  B  ->  ( x  e.  A  \/  -.  x  e.  A
) ) )
7 decidin.ss . . . . . 6  |-  ( ph  ->  A  C_  B )
87ssneld 3014 . . . . 5  |-  ( ph  ->  ( -.  x  e.  B  ->  -.  x  e.  A ) )
9 olc 665 . . . . 5  |-  ( -.  x  e.  A  -> 
( x  e.  A  \/  -.  x  e.  A
) )
108, 9syl6 33 . . . 4  |-  ( ph  ->  ( -.  x  e.  B  ->  ( x  e.  A  \/  -.  x  e.  A )
) )
116, 10jaod 670 . . 3  |-  ( ph  ->  ( ( x  e.  B  \/  -.  x  e.  B )  ->  (
x  e.  A  \/  -.  x  e.  A
) ) )
123, 11syld 44 . 2  |-  ( ph  ->  ( x  e.  C  ->  ( x  e.  A  \/  -.  x  e.  A
) ) )
1312decidr 11053 1  |-  ( ph  ->  A DECIDin  C )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 662    e. wcel 1436    C_ wss 2986   DECIDin wdcin 11050
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-dc 779  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-v 2616  df-in 2992  df-ss 2999  df-dcin 11051
This theorem is referenced by:  sumdc2  11056
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