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Theorem ddifnel 3075
 Description: Double complement under universal class. The hypothesis is one way of expressing the idea that membership in 𝐴 is decidable. 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 3175. (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 262 . . 3 ((𝑥 ∈ V ∧ ¬ 𝑥 ∈ (V ∖ 𝐴)) → 𝑥𝐴)
3 elndif 3068 . . . 4 (𝑥𝐴 → ¬ 𝑥 ∈ (V ∖ 𝐴))
4 vex 2560 . . . 4 𝑥 ∈ V
53, 4jctil 295 . . 3 (𝑥𝐴 → (𝑥 ∈ V ∧ ¬ 𝑥 ∈ (V ∖ 𝐴)))
62, 5impbii 117 . 2 ((𝑥 ∈ V ∧ ¬ 𝑥 ∈ (V ∖ 𝐴)) ↔ 𝑥𝐴)
76difeqri 3064 1 (V ∖ (V ∖ 𝐴)) = 𝐴
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   = wceq 1243   ∈ wcel 1393  Vcvv 2557   ∖ cdif 2914 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-dif 2920 This theorem is referenced by: (None)
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