Theorem ddifnel 3253

Description: Double complement under universal class. The hypothesis corresponds to stability of membership in A, which is weaker than decidability (see dcstab 834). Actually, the conclusion is a characterization of stability of membership in a class (see ddifstab 3254) . Exercise 4.10(s) of [Mendelson] p. 231, but with an additional hypothesis. For a version without a hypothesis, but which only states that A is a subset of _V \ ( _V \ A ), see ddifss 3360. (Contributed by Jim Kingdon, 21-Jul-2018.)

Hypothesis

Ref	Expression
ddifnel.1	|- ( -. x e. ( _V \ A ) -> x e. A )

Assertion

Ref	Expression
ddifnel	|- ( _V \ ( _V \ A ) ) = A

Distinct variable group:   x, A

Proof of Theorem ddifnel

Step	Hyp	Ref	Expression
1		ddifnel.1	. . . 4 |- ( -. x e. ( _V \ A ) -> x e. A )
2	1	adantl 275	. . 3 |- ( ( x e. _V /\ -. x e. ( _V \ A ) ) -> x e. A )
3		elndif 3246	. . . 4 |- ( x e. A -> -. x e. ( _V \ A ) )
4		vex 2729	. . . 4 |- x e. _V
5	3, 4	jctil 310	. . 3 |- ( x e. A -> ( x e. _V /\ -. x e. ( _V \ A ) ) )
6	2, 5	impbii 125	. 2 |- ( ( x e. _V /\ -. x e. ( _V \ A ) ) <-> x e. A )
7	6	difeqri 3242	1 |- ( _V \ ( _V \ A ) ) = A

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   = 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-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)