Theorem ssddif 3171

Description: Double complement and subset. Similar to ddifss 3175 but inside a class B instead of the universal class _V. In classical logic the subset operation on the right hand side could be an equality (that is, A C_ B <-> ( B \ ( B \ A ) ) = A). (Contributed by Jim Kingdon, 24-Jul-2018.)

Assertion

Ref	Expression
ssddif	|- ( A C_ B <-> A C_ ( B \ ( B \ A ) ) )

Proof of Theorem ssddif

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ancr 304	. . . . 5 |- ( ( x e. A -> x e. B ) -> ( x e. A -> ( x e. B /\ x e. A ) ) )
2		simpr 103	. . . . . . . 8 |- ( ( x e. B /\ -. x e. A ) -> -. x e. A )
3	2	con2i 557	. . . . . . 7 |- ( x e. A -> -. ( x e. B /\ -. x e. A ) )
4	3	anim2i 324	. . . . . 6 |- ( ( x e. B /\ x e. A ) -> ( x e. B /\ -. ( x e. B /\ -. x e. A ) ) )
5		eldif 2927	. . . . . . 7 |- ( x e. ( B \ ( B \ A ) ) <-> ( x e. B /\ -. x e. ( B \ A ) ) )
6		eldif 2927	. . . . . . . . 9 |- ( x e. ( B \ A ) <-> ( x e. B /\ -. x e. A ) )
7	6	notbii 594	. . . . . . . 8 |- ( -. x e. ( B \ A ) <-> -. ( x e. B /\ -. x e. A ) )
8	7	anbi2i 430	. . . . . . 7 |- ( ( x e. B /\ -. x e. ( B \ A ) ) <-> ( x e. B /\ -. ( x e. B /\ -. x e. A ) ) )
9	5, 8	bitri 173	. . . . . 6 |- ( x e. ( B \ ( B \ A ) ) <-> ( x e. B /\ -. ( x e. B /\ -. x e. A ) ) )
10	4, 9	sylibr 137	. . . . 5 |- ( ( x e. B /\ x e. A ) -> x e. ( B \ ( B \ A ) ) )
11	1, 10	syl6 29	. . . 4 |- ( ( x e. A -> x e. B ) -> ( x e. A -> x e. ( B \ ( B \ A ) ) ) )
12		eldifi 3066	. . . . 5 |- ( x e. ( B \ ( B \ A ) ) -> x e. B )
13	12	imim2i 12	. . . 4 |- ( ( x e. A -> x e. ( B \ ( B \ A ) ) ) -> ( x e. A -> x e. B ) )
14	11, 13	impbii 117	. . 3 |- ( ( x e. A -> x e. B ) <-> ( x e. A -> x e. ( B \ ( B \ A ) ) ) )
15	14	albii 1359	. 2 |- ( A. x ( x e. A -> x e. B ) <-> A. x ( x e. A -> x e. ( B \ ( B \ A ) ) ) )
16		dfss2 2934	. 2 |- ( A C_ B <-> A. x ( x e. A -> x e. B ) )
17		dfss2 2934	. 2 |- ( A C_ ( B \ ( B \ A ) ) <-> A. x ( x e. A -> x e. ( B \ ( B \ A ) ) ) )
18	15, 16, 17	3bitr4i 201	1 |- ( A C_ B <-> A C_ ( B \ ( B \ A ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   <-> wb 98   A.wal 1241   e. wcel 1393   \ cdif 2914   C_ wss 2917

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-dif 2920  df-in 2924  df-ss 2931

This theorem is referenced by:  ddifss  3175  inssddif  3178