Theorem ddifstab 3259

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 2164	. 2 |- ( ( _V \ ( _V \ A ) ) = A <-> A. x ( x e. ( _V \ ( _V \ A ) ) <-> x e. A ) )
2		eldif 3130	. . . . . . 7 |- ( x e. ( _V \ ( _V \ A ) ) <-> ( x e. _V /\ -. x e. ( _V \ A ) ) )
3		vex 2733	. . . . . . . 8 |- x e. _V
4	3	biantrur 301	. . . . . . 7 |- ( -. x e. ( _V \ A ) <-> ( x e. _V /\ -. x e. ( _V \ A ) ) )
5		eldif 3130	. . . . . . . . 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 663	. . . . . . 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 624	. . . . . . 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 826	. . . 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 1463	. 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 825   A.wal 1346   = wceq 1348   e. wcel 2141   _Vcvv 2730   \ cdif 3118

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-stab 826  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-dif 3123

This theorem is referenced by: (None)