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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 | neeq1 3075 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 ≠ ∅ ↔ 𝐵 ≠ ∅)) | |
2 | soeq2 5488 | . . . . 5 ⊢ (𝑦 = 𝐵 → ( [⊊] Or 𝑦 ↔ [⊊] Or 𝐵)) | |
3 | 1, 2 | anbi12d 630 | . . . 4 ⊢ (𝑦 = 𝐵 → ((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) ↔ (𝐵 ≠ ∅ ∧ [⊊] Or 𝐵))) |
4 | unieq 4838 | . . . . 5 ⊢ (𝑦 = 𝐵 → ∪ 𝑦 = ∪ 𝐵) | |
5 | id 22 | . . . . 5 ⊢ (𝑦 = 𝐵 → 𝑦 = 𝐵) | |
6 | 4, 5 | eleq12d 2904 | . . . 4 ⊢ (𝑦 = 𝐵 → (∪ 𝑦 ∈ 𝑦 ↔ ∪ 𝐵 ∈ 𝐵)) |
7 | 3, 6 | imbi12d 346 | . . 3 ⊢ (𝑦 = 𝐵 → (((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∪ 𝑦 ∈ 𝑦) ↔ ((𝐵 ≠ ∅ ∧ [⊊] Or 𝐵) → ∪ 𝐵 ∈ 𝐵))) |
8 | isfin2 9704 | . . . . 5 ⊢ (𝐴 ∈ FinII → (𝐴 ∈ FinII ↔ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∪ 𝑦 ∈ 𝑦))) | |
9 | 8 | ibi 268 | . . . 4 ⊢ (𝐴 ∈ FinII → ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∪ 𝑦 ∈ 𝑦)) |
10 | 9 | adantr 481 | . . 3 ⊢ ((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝒫 𝐴) → ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∪ 𝑦 ∈ 𝑦)) |
11 | pwexg 5270 | . . . . 5 ⊢ (𝐴 ∈ FinII → 𝒫 𝐴 ∈ V) | |
12 | elpw2g 5238 | . . . . 5 ⊢ (𝒫 𝐴 ∈ V → (𝐵 ∈ 𝒫 𝒫 𝐴 ↔ 𝐵 ⊆ 𝒫 𝐴)) | |
13 | 11, 12 | syl 17 | . . . 4 ⊢ (𝐴 ∈ FinII → (𝐵 ∈ 𝒫 𝒫 𝐴 ↔ 𝐵 ⊆ 𝒫 𝐴)) |
14 | 13 | biimpar 478 | . . 3 ⊢ ((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝒫 𝐴) → 𝐵 ∈ 𝒫 𝒫 𝐴) |
15 | 7, 10, 14 | rspcdva 3622 | . 2 ⊢ ((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝒫 𝐴) → ((𝐵 ≠ ∅ ∧ [⊊] Or 𝐵) → ∪ 𝐵 ∈ 𝐵)) |
16 | 15 | imp 407 | 1 ⊢ (((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝒫 𝐴) ∧ (𝐵 ≠ ∅ ∧ [⊊] Or 𝐵)) → ∪ 𝐵 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ∀wral 3135 Vcvv 3492 ⊆ wss 3933 ∅c0 4288 𝒫 cpw 4535 ∪ cuni 4830 Or wor 5466 [⊊] crpss 7437 FinIIcfin2 9689 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-pow 5257 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-in 3940 df-ss 3949 df-pw 4537 df-uni 4831 df-po 5467 df-so 5468 df-fin2 9696 |
This theorem is referenced by: fin2i2 9728 ssfin2 9730 enfin2i 9731 fin1a2lem13 9822 |
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