Theorem ddifstab 3116

Description: A class is equal to its double complement if and only if it is stable (that is, membership in it is a stable property). (Contributed by BJ, 12-Dec-2021.)

Assertion

Ref	Expression
ddifstab	|- ( ( _V \ ( _V \ A ) ) = A <-> A. xSTAB x e. A )

Distinct variable group:   x, A

Proof of Theorem ddifstab

Step	Hyp	Ref	Expression
1		dfcleq 2077	. 2 |- ( ( _V \ ( _V \ A ) ) = A <-> A. x ( x e. ( _V \ ( _V \ A ) ) <-> x e. A ) )
2		eldif 2993	. . . . . . 7 |- ( x e. ( _V \ ( _V \ A ) ) <-> ( x e. _V /\ -. x e. ( _V \ A ) ) )
3		vex 2615	. . . . . . . 8 |- x e. _V
4	3	biantrur 297	. . . . . . 7 |- ( -. x e. ( _V \ A ) <-> ( x e. _V /\ -. x e. ( _V \ A ) ) )
5		eldif 2993	. . . . . . . . 9 |- ( x e. ( _V \ A ) <-> ( x e. _V /\ -. x e. A ) )
6	3	biantrur 297	. . . . . . . . 9 |- ( -. x e. A <-> ( x e. _V /\ -. x e. A ) )
7	5, 6	bitr4i 185	. . . . . . . 8 |- ( x e. ( _V \ A ) <-> -. x e. A )
8	7	notbii 627	. . . . . . 7 |- ( -. x e. ( _V \ A ) <-> -. -. x e. A )
9	2, 4, 8	3bitr2i 206	. . . . . 6 |- ( x e. ( _V \ ( _V \ A ) ) <-> -. -. x e. A )
10	9	bibi1i 226	. . . . 5 |- ( ( x e. ( _V \ ( _V \ A ) ) <-> x e. A ) <-> ( -. -. x e. A <-> x e. A ) )
11		bi1 116	. . . . . 6 |- ( ( -. -. x e. A <-> x e. A ) -> ( -. -. x e. A -> x e. A ) )
12		id 19	. . . . . . 7 |- ( ( -. -. x e. A -> x e. A ) -> ( -. -. x e. A -> x e. A ) )
13		notnot 592	. . . . . . 7 |- ( x e. A -> -. -. x e. A )
14	12, 13	impbid1 140	. . . . . 6 |- ( ( -. -. x e. A -> x e. A ) -> ( -. -. x e. A <-> x e. A ) )
15	11, 14	impbii 124	. . . . 5 |- ( ( -. -. x e. A <-> x e. A ) <-> ( -. -. x e. A -> x e. A ) )
16	10, 15	bitri 182	. . . 4 |- ( ( x e. ( _V \ ( _V \ A ) ) <-> x e. A ) <-> ( -. -. x e. A -> x e. A ) )
17		df-stab 774	. . . 4 |- (STAB x e. A <-> ( -. -. x e. A -> x e. A ) )
18	16, 17	bitr4i 185	. . 3 |- ( ( x e. ( _V \ ( _V \ A ) ) <-> x e. A ) <-> STAB x e. A )
19	18	albii 1400	. 2 |- ( A. x ( x e. ( _V \ ( _V \ A ) ) <-> x e. A ) <-> A. xSTAB x e. A )
20	1, 19	bitri 182	1 |- ( ( _V \ ( _V \ A ) ) = A <-> A. xSTAB x e. A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   <-> wb 103  STAB wstab 773   A.wal 1283   = wceq 1285   e. wcel 1434   _Vcvv 2612   \ cdif 2981

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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-stab 774  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614  df-dif 2986

This theorem is referenced by: (None)