Theorem ddifstab 3172

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 2107	. 2 |- ( ( _V \ ( _V \ A ) ) = A <-> A. x ( x e. ( _V \ ( _V \ A ) ) <-> x e. A ) )
2		eldif 3044	. . . . . . 7 |- ( x e. ( _V \ ( _V \ A ) ) <-> ( x e. _V /\ -. x e. ( _V \ A ) ) )
3		vex 2658	. . . . . . . 8 |- x e. _V
4	3	biantrur 299	. . . . . . 7 |- ( -. x e. ( _V \ A ) <-> ( x e. _V /\ -. x e. ( _V \ A ) ) )
5		eldif 3044	. . . . . . . . 9 |- ( x e. ( _V \ A ) <-> ( x e. _V /\ -. x e. A ) )
6	3	biantrur 299	. . . . . . . . 9 |- ( -. x e. A <-> ( x e. _V /\ -. x e. A ) )
7	5, 6	bitr4i 186	. . . . . . . 8 |- ( x e. ( _V \ A ) <-> -. x e. A )
8	7	notbii 640	. . . . . . 7 |- ( -. x e. ( _V \ A ) <-> -. -. x e. A )
9	2, 4, 8	3bitr2i 207	. . . . . 6 |- ( x e. ( _V \ ( _V \ A ) ) <-> -. -. x e. A )
10	9	bibi1i 227	. . . . 5 |- ( ( x e. ( _V \ ( _V \ A ) ) <-> x e. A ) <-> ( -. -. x e. A <-> x e. A ) )
11		bi1 117	. . . . . 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 601	. . . . . . 7 |- ( x e. A -> -. -. x e. A )
14	12, 13	impbid1 141	. . . . . 6 |- ( ( -. -. x e. A -> x e. A ) -> ( -. -. x e. A <-> x e. A ) )
15	11, 14	impbii 125	. . . . 5 |- ( ( -. -. x e. A <-> x e. A ) <-> ( -. -. x e. A -> x e. A ) )
16	10, 15	bitri 183	. . . 4 |- ( ( x e. ( _V \ ( _V \ A ) ) <-> x e. A ) <-> ( -. -. x e. A -> x e. A ) )
17		df-stab 799	. . . 4 |- (STAB x e. A <-> ( -. -. x e. A -> x e. A ) )
18	16, 17	bitr4i 186	. . 3 |- ( ( x e. ( _V \ ( _V \ A ) ) <-> x e. A ) <-> STAB x e. A )
19	18	albii 1427	. 2 |- ( A. x ( x e. ( _V \ ( _V \ A ) ) <-> x e. A ) <-> A. xSTAB x e. A )
20	1, 19	bitri 183	1 |- ( ( _V \ ( _V \ A ) ) = A <-> A. xSTAB x e. A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104  STAB wstab 798   A.wal 1310   = wceq 1312   e. wcel 1461   _Vcvv 2655   \ cdif 3032

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095

This theorem depends on definitions:  df-bi 116  df-stab 799  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-v 2657  df-dif 3037

This theorem is referenced by: (None)