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.

Step	Hyp	Ref	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 ) ) )
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 11695	. . . . 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 3027	. . . . 5 |- ( ph -> ( -. x e. B -> -. x e. A ) )
9		olc 667	. . . . 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 672	. . 3 |- ( ph -> ( ( x e. B \/ -. x e. B ) -> ( x e. A \/ -. x e. A ) ) )
12	3, 11	syld 44	. 2 |- ( ph -> ( x e. C -> ( x e. A \/ -. x e. A ) ) )
13	12	decidr 11696	1 |- ( ph -> A DECIDin C )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 664   e. wcel 1438   C_ wss 2999   DECID_in 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