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Theorem decidin 15010
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 15008 . . . 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 15008 . . . . 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 3172 . . . . 5  |-  ( ph  ->  ( -.  x  e.  B  ->  -.  x  e.  A ) )
9 olc 712 . . . . 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 718 . . 3  |-  ( ph  ->  ( ( x  e.  B  \/  -.  x  e.  B )  ->  (
x  e.  A  \/  -.  x  e.  A
) ) )
123, 11syld 45 . 2  |-  ( ph  ->  ( x  e.  C  ->  ( x  e.  A  \/  -.  x  e.  A
) ) )
1312decidr 15009 1  |-  ( ph  ->  A DECIDin  C )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 709    e. wcel 2160    C_ wss 3144   DECIDin wdcin 15006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-dc 836  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-v 2754  df-in 3150  df-ss 3157  df-dcin 15007
This theorem is referenced by:  sumdc2  15012
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