Theorem decidin 14107

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 14105	. . . 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 14105	. . . . 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 3155	. . . . 5 |- ( ph -> ( -. x e. B -> -. x e. A ) )
9		olc 711	. . . . 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 717	. . 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 14106	1 |- ( ph -> A DECIDin C )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 708   e. wcel 2146   C_ wss 3127   DECID_in wdcin 14103

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-dc 835  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-v 2737  df-in 3133  df-ss 3140  df-dcin 14104

This theorem is referenced by:  sumdc2  14109