Theorem ddifnel 3212

Description: Double complement under universal class. The hypothesis corresponds to stability of membership in A, which is weaker than decidability (see dcstab 830). Actually, the conclusion is a characterization of stability of membership in a class (see ddifstab 3213) . 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 3319. (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 3205	. . . 4 |- ( x e. A -> -. x e. ( _V \ A ) )
4		vex 2692	. . . 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 3201	1 |- ( _V \ ( _V \ A ) ) = A

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   = wceq 1332   e. wcel 1481   _Vcvv 2689   \ cdif 3073

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-dif 3078

This theorem is referenced by: (None)