Theorem ddifnel 3294

Description: Double complement under universal class. The hypothesis corresponds to stability of membership in 𝐴, which is weaker than decidability (see dcstab 845). Actually, the conclusion is a characterization of stability of membership in a class (see ddifstab 3295) . 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 3401. (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 3287	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ (V ∖ 𝐴))
4		vex 2766	. . . 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 3283	1 ⊢ (V ∖ (V ∖ 𝐴)) = 𝐴

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2167  Vcvv 2763   ∖ cdif 3154

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-dif 3159

This theorem is referenced by: (None)