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Theorem ddifnel 3281
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 3282) . 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 3388. (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 277 . . 3 ((𝑥 ∈ V ∧ ¬ 𝑥 ∈ (V ∖ 𝐴)) → 𝑥𝐴)
3 elndif 3274 . . . 4 (𝑥𝐴 → ¬ 𝑥 ∈ (V ∖ 𝐴))
4 vex 2755 . . . 4 𝑥 ∈ V
53, 4jctil 312 . . 3 (𝑥𝐴 → (𝑥 ∈ V ∧ ¬ 𝑥 ∈ (V ∖ 𝐴)))
62, 5impbii 126 . 2 ((𝑥 ∈ V ∧ ¬ 𝑥 ∈ (V ∖ 𝐴)) ↔ 𝑥𝐴)
76difeqri 3270 1 (V ∖ (V ∖ 𝐴)) = 𝐴
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104   = wceq 1364  wcel 2160  Vcvv 2752  cdif 3141
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-dif 3146
This theorem is referenced by: (None)
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