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Theorem ddifnel 3117
Description: Double complement under universal class. The hypothesis corresponds to stability of membership in 𝐴, which is weaker than decidability (see dcimpstab 788). Actually, the conclusion is a characterization of stability of membership in a class (see ddifstab 3118) . 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 3223. (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 271 . . 3 ((𝑥 ∈ V ∧ ¬ 𝑥 ∈ (V ∖ 𝐴)) → 𝑥𝐴)
3 elndif 3110 . . . 4 (𝑥𝐴 → ¬ 𝑥 ∈ (V ∖ 𝐴))
4 vex 2617 . . . 4 𝑥 ∈ V
53, 4jctil 305 . . 3 (𝑥𝐴 → (𝑥 ∈ V ∧ ¬ 𝑥 ∈ (V ∖ 𝐴)))
62, 5impbii 124 . 2 ((𝑥 ∈ V ∧ ¬ 𝑥 ∈ (V ∖ 𝐴)) ↔ 𝑥𝐴)
76difeqri 3106 1 (V ∖ (V ∖ 𝐴)) = 𝐴
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102   = wceq 1287  wcel 1436  Vcvv 2614  cdif 2983
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2616  df-dif 2988
This theorem is referenced by: (None)
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