Theorem ddifstab 3254

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 2159	. 2 |- ( ( _V \ ( _V \ A ) ) = A <-> A. x ( x e. ( _V \ ( _V \ A ) ) <-> x e. A ) )
2		eldif 3125	. . . . . . 7 |- ( x e. ( _V \ ( _V \ A ) ) <-> ( x e. _V /\ -. x e. ( _V \ A ) ) )
3		vex 2729	. . . . . . . 8 |- x e. _V
4	3	biantrur 301	. . . . . . 7 |- ( -. x e. ( _V \ A ) <-> ( x e. _V /\ -. x e. ( _V \ A ) ) )
5		eldif 3125	. . . . . . . . 9 |- ( x e. ( _V \ A ) <-> ( x e. _V /\ -. x e. A ) )
6	3	biantrur 301	. . . . . . . . 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 658	. . . . . . 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		biimp 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 619	. . . . . . 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 821	. . . 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 1458	. 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 820   A.wal 1341   = wceq 1343   e. wcel 2136   _Vcvv 2726   \ cdif 3113

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-stab 821  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-dif 3118

This theorem is referenced by: (None)