Theorem ddifnel 3131

Description: Double complement under universal class. The hypothesis corresponds to stability of membership in A, which is weaker than decidability (see dcimpstab 790). Actually, the conclusion is a characterization of stability of membership in a class (see ddifstab 3132) . 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 3237. (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 271	. . 3 |- ( ( x e. _V /\ -. x e. ( _V \ A ) ) -> x e. A )
3		elndif 3124	. . . 4 |- ( x e. A -> -. x e. ( _V \ A ) )
4		vex 2622	. . . 4 |- x e. _V
5	3, 4	jctil 305	. . 3 |- ( x e. A -> ( x e. _V /\ -. x e. ( _V \ A ) ) )
6	2, 5	impbii 124	. 2 |- ( ( x e. _V /\ -. x e. ( _V \ A ) ) <-> x e. A )
7	6	difeqri 3120	1 |- ( _V \ ( _V \ A ) ) = A

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   = wceq 1289   e. wcel 1438   _Vcvv 2619   \ cdif 2996

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-dif 3001

This theorem is referenced by: (None)