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Mirrors > Home > ILE Home > Th. List > ddifnel | GIF version |
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.) |
Ref | Expression |
---|---|
ddifnel.1 | ⊢ (¬ 𝑥 ∈ (V ∖ 𝐴) → 𝑥 ∈ 𝐴) |
Ref | Expression |
---|---|
ddifnel | ⊢ (V ∖ (V ∖ 𝐴)) = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ddifnel.1 | . . . 4 ⊢ (¬ 𝑥 ∈ (V ∖ 𝐴) → 𝑥 ∈ 𝐴) | |
2 | 1 | adantl 277 | . . 3 ⊢ ((𝑥 ∈ V ∧ ¬ 𝑥 ∈ (V ∖ 𝐴)) → 𝑥 ∈ 𝐴) |
3 | elndif 3274 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ (V ∖ 𝐴)) | |
4 | vex 2755 | . . . 4 ⊢ 𝑥 ∈ V | |
5 | 3, 4 | jctil 312 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (𝑥 ∈ V ∧ ¬ 𝑥 ∈ (V ∖ 𝐴))) |
6 | 2, 5 | impbii 126 | . 2 ⊢ ((𝑥 ∈ V ∧ ¬ 𝑥 ∈ (V ∖ 𝐴)) ↔ 𝑥 ∈ 𝐴) |
7 | 6 | difeqri 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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