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.

Step	Hyp	Ref	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 ) ) )
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 15008	. . . . 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 3172	. . . . 5 |- ( ph -> ( -. x e. B -> -. x e. A ) )
9		olc 712	. . . . 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 718	. . 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 15009	1 |- ( ph -> A DECIDin C )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 709   e. wcel 2160   C_ wss 3144   DECID_in 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