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Theorem ddifnel 3146
 Description: Double complement under universal class. The hypothesis corresponds to stability of membership in 𝐴, which is weaker than decidability (see dcimpstab 793). Actually, the conclusion is a characterization of stability of membership in a class (see ddifstab 3147) . 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 3253. (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 272 . . 3 ((𝑥 ∈ V ∧ ¬ 𝑥 ∈ (V ∖ 𝐴)) → 𝑥𝐴)
3 elndif 3139 . . . 4 (𝑥𝐴 → ¬ 𝑥 ∈ (V ∖ 𝐴))
4 vex 2636 . . . 4 𝑥 ∈ V
53, 4jctil 306 . . 3 (𝑥𝐴 → (𝑥 ∈ V ∧ ¬ 𝑥 ∈ (V ∖ 𝐴)))
62, 5impbii 125 . 2 ((𝑥 ∈ V ∧ ¬ 𝑥 ∈ (V ∖ 𝐴)) ↔ 𝑥𝐴)
76difeqri 3135 1 (V ∖ (V ∖ 𝐴)) = 𝐴
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   = wceq 1296   ∈ wcel 1445  Vcvv 2633   ∖ cdif 3010 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077 This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-v 2635  df-dif 3015 This theorem is referenced by: (None)
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