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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > isldsys | Structured version Visualization version GIF version |
Description: The property of being a lambda-system or Dynkin system. Lambda-systems contain the empty set, are closed under complement, and closed under countable disjoint union. (Contributed by Thierry Arnoux, 13-Jun-2020.) |
Ref | Expression |
---|---|
isldsys.l | ⊢ 𝐿 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑠))} |
Ref | Expression |
---|---|
isldsys | ⊢ (𝑆 ∈ 𝐿 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ (∅ ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑂 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑆)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq2 2814 | . . 3 ⊢ (𝑠 = 𝑆 → (∅ ∈ 𝑠 ↔ ∅ ∈ 𝑆)) | |
2 | eleq2 2814 | . . . 4 ⊢ (𝑠 = 𝑆 → ((𝑂 ∖ 𝑥) ∈ 𝑠 ↔ (𝑂 ∖ 𝑥) ∈ 𝑆)) | |
3 | 2 | raleqbi1dv 3325 | . . 3 ⊢ (𝑠 = 𝑆 → (∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ↔ ∀𝑥 ∈ 𝑆 (𝑂 ∖ 𝑥) ∈ 𝑆)) |
4 | pweq 4609 | . . . 4 ⊢ (𝑠 = 𝑆 → 𝒫 𝑠 = 𝒫 𝑆) | |
5 | eleq2 2814 | . . . . 5 ⊢ (𝑠 = 𝑆 → (∪ 𝑥 ∈ 𝑠 ↔ ∪ 𝑥 ∈ 𝑆)) | |
6 | 5 | imbi2d 340 | . . . 4 ⊢ (𝑠 = 𝑆 → (((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑠) ↔ ((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑆))) |
7 | 4, 6 | raleqbidv 3334 | . . 3 ⊢ (𝑠 = 𝑆 → (∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑠) ↔ ∀𝑥 ∈ 𝒫 𝑆((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑆))) |
8 | 1, 3, 7 | 3anbi123d 1432 | . 2 ⊢ (𝑠 = 𝑆 → ((∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑠)) ↔ (∅ ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑂 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑆)))) |
9 | isldsys.l | . 2 ⊢ 𝐿 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑠))} | |
10 | 8, 9 | elrab2 3679 | 1 ⊢ (𝑆 ∈ 𝐿 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ (∅ ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑂 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑆)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3053 {crab 3424 ∖ cdif 3938 ∅c0 4315 𝒫 cpw 4595 ∪ cuni 4900 Disj wdisj 5104 class class class wbr 5139 ωcom 7849 ≼ cdom 8934 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-3an 1086 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-in 3948 df-ss 3958 df-pw 4597 |
This theorem is referenced by: pwldsys 33674 unelldsys 33675 sigaldsys 33676 ldsysgenld 33677 sigapildsyslem 33678 sigapildsys 33679 ldgenpisyslem1 33680 |
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