Theorem ddifnel 3132

Description: Double complement under universal class. The hypothesis corresponds to stability of membership in A, which is weaker than decidability (see dcimpstab 791). Actually, the conclusion is a characterization of stability of membership in a class (see ddifstab 3133) . 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 3238. (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 272	. . 3 |- ( ( x e. _V /\ -. x e. ( _V \ A ) ) -> x e. A )
3		elndif 3125	. . . 4 |- ( x e. A -> -. x e. ( _V \ A ) )
4		vex 2623	. . . 4 |- x e. _V
5	3, 4	jctil 306	. . 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 3121	1 |- ( _V \ ( _V \ A ) ) = A

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   = wceq 1290   e. wcel 1439   _Vcvv 2620   \ cdif 2997

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 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-dif 3002

This theorem is referenced by: (None)