Theorem ddifnel 3264

Description: Double complement under universal class. The hypothesis corresponds to stability of membership in A, which is weaker than decidability (see dcstab 844). Actually, the conclusion is a characterization of stability of membership in a class (see ddifstab 3265) . 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 3371. (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 277	. . 3 |- ( ( x e. _V /\ -. x e. ( _V \ A ) ) -> x e. A )
3		elndif 3257	. . . 4 |- ( x e. A -> -. x e. ( _V \ A ) )
4		vex 2738	. . . 4 |- x e. _V
5	3, 4	jctil 312	. . 3 |- ( x e. A -> ( x e. _V /\ -. x e. ( _V \ A ) ) )
6	2, 5	impbii 126	. 2 |- ( ( x e. _V /\ -. x e. ( _V \ A ) ) <-> x e. A )
7	6	difeqri 3253	1 |- ( _V \ ( _V \ A ) ) = A

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2146   _Vcvv 2735   \ cdif 3124

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-v 2737  df-dif 3129

This theorem is referenced by: (None)