Theorem decidin 13175

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.

Step	Hyp	Ref	Expression
1		decidin.b	. . . 4 |- ( ph -> B DECIDin C )
2		decidi 13173	. . . 4 |- ( B DECIDin C -> ( x e. C -> ( x e. B \/ -. x e. B ) ) )
3	1, 2	syl 14	. . 3 |- ( ph -> ( x e. C -> ( x e. B \/ -. x e. B ) ) )
4		decidin.a	. . . . 5 |- ( ph -> A DECIDin B )
5		decidi 13173	. . . . 5 |- ( A DECIDin B -> ( x e. B -> ( x e. A \/ -. x e. A ) ) )
6	4, 5	syl 14	. . . 4 |- ( ph -> ( x e. B -> ( x e. A \/ -. x e. A ) ) )
7		decidin.ss	. . . . . 6 |- ( ph -> A C_ B )
8	7	ssneld 3104	. . . . 5 |- ( ph -> ( -. x e. B -> -. x e. A ) )
9		olc 701	. . . . 5 |- ( -. x e. A -> ( x e. A \/ -. x e. A ) )
10	8, 9	syl6 33	. . . 4 |- ( ph -> ( -. x e. B -> ( x e. A \/ -. x e. A ) ) )
11	6, 10	jaod 707	. . 3 |- ( ph -> ( ( x e. B \/ -. x e. B ) -> ( x e. A \/ -. x e. A ) ) )
12	3, 11	syld 45	. 2 |- ( ph -> ( x e. C -> ( x e. A \/ -. x e. A ) ) )
13	12	decidr 13174	1 |- ( ph -> A DECIDin C )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 698   e. wcel 1481   C_ wss 3076   DECID_in wdcin 13171

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-dc 821  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-v 2691  df-in 3082  df-ss 3089  df-dcin 13172

This theorem is referenced by:  sumdc2  13177