Theorem ddifnel 3268

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 3269) . 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 3375. (Contributed by Jim Kingdon, 21-Jul-2018.)

Hypothesis

Ref	Expression
ddifnel.1	⊢ (¬ 𝑥 ∈ (V ∖ 𝐴) → 𝑥 ∈ 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 ∈ (V ∖ 𝐴) → 𝑥 ∈ 𝐴)
2	1	adantl 277	. . 3 ⊢ ((𝑥 ∈ V ∧ ¬ 𝑥 ∈ (V ∖ 𝐴)) → 𝑥 ∈ 𝐴)
3		elndif 3261	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ (V ∖ 𝐴))
4		vex 2742	. . . 4 ⊢ 𝑥 ∈ V
5	3, 4	jctil 312	. . 3 ⊢ (𝑥 ∈ 𝐴 → (𝑥 ∈ V ∧ ¬ 𝑥 ∈ (V ∖ 𝐴)))
6	2, 5	impbii 126	. 2 ⊢ ((𝑥 ∈ V ∧ ¬ 𝑥 ∈ (V ∖ 𝐴)) ↔ 𝑥 ∈ 𝐴)
7	6	difeqri 3257	1 ⊢ (V ∖ (V ∖ 𝐴)) = 𝐴

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   = wceq 1353   ∈ wcel 2148  Vcvv 2739   ∖ cdif 3128

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-dif 3133

This theorem is referenced by: (None)