Theorem ssddif 3406

Description: Double complement and subset. Similar to ddifss 3410 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 321	. . . . 5 |- ( ( x e. A -> x e. B ) -> ( x e. A -> ( x e. B /\ x e. A ) ) )
2		simpr 110	. . . . . . . 8 |- ( ( x e. B /\ -. x e. A ) -> -. x e. A )
3	2	con2i 628	. . . . . . 7 |- ( x e. A -> -. ( x e. B /\ -. x e. A ) )
4	3	anim2i 342	. . . . . 6 |- ( ( x e. B /\ x e. A ) -> ( x e. B /\ -. ( x e. B /\ -. x e. A ) ) )
5		eldif 3174	. . . . . . 7 |- ( x e. ( B \ ( B \ A ) ) <-> ( x e. B /\ -. x e. ( B \ A ) ) )
6		eldif 3174	. . . . . . . . 9 |- ( x e. ( B \ A ) <-> ( x e. B /\ -. x e. A ) )
7	6	notbii 669	. . . . . . . 8 |- ( -. x e. ( B \ A ) <-> -. ( x e. B /\ -. x e. A ) )
8	7	anbi2i 457	. . . . . . 7 |- ( ( x e. B /\ -. x e. ( B \ A ) ) <-> ( x e. B /\ -. ( x e. B /\ -. x e. A ) ) )
9	5, 8	bitri 184	. . . . . 6 |- ( x e. ( B \ ( B \ A ) ) <-> ( x e. B /\ -. ( x e. B /\ -. x e. A ) ) )
10	4, 9	sylibr 134	. . . . 5 |- ( ( x e. B /\ x e. A ) -> x e. ( B \ ( B \ A ) ) )
11	1, 10	syl6 33	. . . 4 |- ( ( x e. A -> x e. B ) -> ( x e. A -> x e. ( B \ ( B \ A ) ) ) )
12		eldifi 3294	. . . . 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 126	. . 3 |- ( ( x e. A -> x e. B ) <-> ( x e. A -> x e. ( B \ ( B \ A ) ) ) )
15	14	albii 1492	. 2 |- ( A. x ( x e. A -> x e. B ) <-> A. x ( x e. A -> x e. ( B \ ( B \ A ) ) ) )
16		ssalel 3180	. 2 |- ( A C_ B <-> A. x ( x e. A -> x e. B ) )
17		ssalel 3180	. 2 |- ( A C_ ( B \ ( B \ A ) ) <-> A. x ( x e. A -> x e. ( B \ ( B \ A ) ) ) )
18	15, 16, 17	3bitr4i 212	1 |- ( A C_ B <-> A C_ ( B \ ( B \ A ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1370   e. wcel 2175   \ cdif 3162   C_ wss 3165

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-dif 3167  df-in 3171  df-ss 3178

This theorem is referenced by:  ddifss  3410  inssddif  3413