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Mirrors > Home > MPE Home > Th. List > fin2i | Structured version Visualization version GIF version |
Description: Property of a II-finite set. (Contributed by Stefan O'Rear, 16-May-2015.) |
Ref | Expression |
---|---|
fin2i | ⊢ (((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝒫 𝐴) ∧ (𝐵 ≠ ∅ ∧ [⊊] Or 𝐵)) → ∪ 𝐵 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwexg 4776 | . . . . 5 ⊢ (𝐴 ∈ FinII → 𝒫 𝐴 ∈ V) | |
2 | elpw2g 4754 | . . . . 5 ⊢ (𝒫 𝐴 ∈ V → (𝐵 ∈ 𝒫 𝒫 𝐴 ↔ 𝐵 ⊆ 𝒫 𝐴)) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝐴 ∈ FinII → (𝐵 ∈ 𝒫 𝒫 𝐴 ↔ 𝐵 ⊆ 𝒫 𝐴)) |
4 | 3 | biimpar 501 | . . 3 ⊢ ((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝒫 𝐴) → 𝐵 ∈ 𝒫 𝒫 𝐴) |
5 | isfin2 8999 | . . . . 5 ⊢ (𝐴 ∈ FinII → (𝐴 ∈ FinII ↔ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∪ 𝑦 ∈ 𝑦))) | |
6 | 5 | ibi 255 | . . . 4 ⊢ (𝐴 ∈ FinII → ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∪ 𝑦 ∈ 𝑦)) |
7 | 6 | adantr 480 | . . 3 ⊢ ((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝒫 𝐴) → ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∪ 𝑦 ∈ 𝑦)) |
8 | neeq1 2844 | . . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑦 ≠ ∅ ↔ 𝐵 ≠ ∅)) | |
9 | soeq2 4979 | . . . . . 6 ⊢ (𝑦 = 𝐵 → ( [⊊] Or 𝑦 ↔ [⊊] Or 𝐵)) | |
10 | 8, 9 | anbi12d 743 | . . . . 5 ⊢ (𝑦 = 𝐵 → ((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) ↔ (𝐵 ≠ ∅ ∧ [⊊] Or 𝐵))) |
11 | unieq 4380 | . . . . . 6 ⊢ (𝑦 = 𝐵 → ∪ 𝑦 = ∪ 𝐵) | |
12 | id 22 | . . . . . 6 ⊢ (𝑦 = 𝐵 → 𝑦 = 𝐵) | |
13 | 11, 12 | eleq12d 2682 | . . . . 5 ⊢ (𝑦 = 𝐵 → (∪ 𝑦 ∈ 𝑦 ↔ ∪ 𝐵 ∈ 𝐵)) |
14 | 10, 13 | imbi12d 333 | . . . 4 ⊢ (𝑦 = 𝐵 → (((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∪ 𝑦 ∈ 𝑦) ↔ ((𝐵 ≠ ∅ ∧ [⊊] Or 𝐵) → ∪ 𝐵 ∈ 𝐵))) |
15 | 14 | rspcv 3278 | . . 3 ⊢ (𝐵 ∈ 𝒫 𝒫 𝐴 → (∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∪ 𝑦 ∈ 𝑦) → ((𝐵 ≠ ∅ ∧ [⊊] Or 𝐵) → ∪ 𝐵 ∈ 𝐵))) |
16 | 4, 7, 15 | sylc 63 | . 2 ⊢ ((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝒫 𝐴) → ((𝐵 ≠ ∅ ∧ [⊊] Or 𝐵) → ∪ 𝐵 ∈ 𝐵)) |
17 | 16 | imp 444 | 1 ⊢ (((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝒫 𝐴) ∧ (𝐵 ≠ ∅ ∧ [⊊] Or 𝐵)) → ∪ 𝐵 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 Vcvv 3173 ⊆ wss 3540 ∅c0 3874 𝒫 cpw 4108 ∪ cuni 4372 Or wor 4958 [⊊] crpss 6834 FinIIcfin2 8984 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-pow 4769 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-v 3175 df-in 3547 df-ss 3554 df-pw 4110 df-uni 4373 df-po 4959 df-so 4960 df-fin2 8991 |
This theorem is referenced by: fin2i2 9023 ssfin2 9025 enfin2i 9026 fin1a2lem13 9117 |
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