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Theorem decidin 11697
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 11695 . . . 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 11695 . . . . 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 3027 . . . . 5  |-  ( ph  ->  ( -.  x  e.  B  ->  -.  x  e.  A ) )
9 olc 667 . . . . 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 672 . . 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 11696 1  |-  ( ph  ->  A DECIDin  C )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 664    e. wcel 1438    C_ wss 2999   DECIDin wdcin 11693
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-dc 781  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-in 3005  df-ss 3012  df-dcin 11694
This theorem is referenced by:  sumdc2  11699
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