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Theorem ddifnel 3207
 Description: Double complement under universal class. The hypothesis corresponds to stability of membership in 𝐴, which is weaker than decidability (see dcstab 829). Actually, the conclusion is a characterization of stability of membership in a class (see ddifstab 3208) . 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 3314. (Contributed by Jim Kingdon, 21-Jul-2018.)
Hypothesis
Ref Expression
ddifnel.1 𝑥 ∈ (V ∖ 𝐴) → 𝑥𝐴)
Assertion
Ref Expression
ddifnel (V ∖ (V ∖ 𝐴)) = 𝐴
Distinct variable group:   𝑥,𝐴

Proof of Theorem ddifnel
StepHypRef Expression
1 ddifnel.1 . . . 4 𝑥 ∈ (V ∖ 𝐴) → 𝑥𝐴)
21adantl 275 . . 3 ((𝑥 ∈ V ∧ ¬ 𝑥 ∈ (V ∖ 𝐴)) → 𝑥𝐴)
3 elndif 3200 . . . 4 (𝑥𝐴 → ¬ 𝑥 ∈ (V ∖ 𝐴))
4 vex 2689 . . . 4 𝑥 ∈ V
53, 4jctil 310 . . 3 (𝑥𝐴 → (𝑥 ∈ V ∧ ¬ 𝑥 ∈ (V ∖ 𝐴)))
62, 5impbii 125 . 2 ((𝑥 ∈ V ∧ ¬ 𝑥 ∈ (V ∖ 𝐴)) ↔ 𝑥𝐴)
76difeqri 3196 1 (V ∖ (V ∖ 𝐴)) = 𝐴
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   = wceq 1331   ∈ wcel 1480  Vcvv 2686   ∖ cdif 3068 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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-dif 3073 This theorem is referenced by: (None)
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