Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 3078 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 ≠ ∅ ↔ 𝐵 ≠ ∅)) | |
2 | soeq2 5495 | . . . . 5 ⊢ (𝑦 = 𝐵 → ( [⊊] Or 𝑦 ↔ [⊊] Or 𝐵)) | |
3 | 1, 2 | anbi12d 632 | . . . 4 ⊢ (𝑦 = 𝐵 → ((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) ↔ (𝐵 ≠ ∅ ∧ [⊊] Or 𝐵))) |
4 | unieq 4849 | . . . . 5 ⊢ (𝑦 = 𝐵 → ∪ 𝑦 = ∪ 𝐵) | |
5 | id 22 | . . . . 5 ⊢ (𝑦 = 𝐵 → 𝑦 = 𝐵) | |
6 | 4, 5 | eleq12d 2907 | . . . 4 ⊢ (𝑦 = 𝐵 → (∪ 𝑦 ∈ 𝑦 ↔ ∪ 𝐵 ∈ 𝐵)) |
7 | 3, 6 | imbi12d 347 | . . 3 ⊢ (𝑦 = 𝐵 → (((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∪ 𝑦 ∈ 𝑦) ↔ ((𝐵 ≠ ∅ ∧ [⊊] Or 𝐵) → ∪ 𝐵 ∈ 𝐵))) |
8 | isfin2 9716 | . . . . 5 ⊢ (𝐴 ∈ FinII → (𝐴 ∈ FinII ↔ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∪ 𝑦 ∈ 𝑦))) | |
9 | 8 | ibi 269 | . . . 4 ⊢ (𝐴 ∈ FinII → ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∪ 𝑦 ∈ 𝑦)) |
10 | 9 | adantr 483 | . . 3 ⊢ ((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝒫 𝐴) → ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∪ 𝑦 ∈ 𝑦)) |
11 | pwexg 5279 | . . . . 5 ⊢ (𝐴 ∈ FinII → 𝒫 𝐴 ∈ V) | |
12 | elpw2g 5247 | . . . . 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 480 | . . 3 ⊢ ((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝒫 𝐴) → 𝐵 ∈ 𝒫 𝒫 𝐴) |
15 | 7, 10, 14 | rspcdva 3625 | . 2 ⊢ ((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝒫 𝐴) → ((𝐵 ≠ ∅ ∧ [⊊] Or 𝐵) → ∪ 𝐵 ∈ 𝐵)) |
16 | 15 | imp 409 | 1 ⊢ (((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝒫 𝐴) ∧ (𝐵 ≠ ∅ ∧ [⊊] Or 𝐵)) → ∪ 𝐵 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∀wral 3138 Vcvv 3494 ⊆ wss 3936 ∅c0 4291 𝒫 cpw 4539 ∪ cuni 4838 Or wor 5473 [⊊] crpss 7448 FinIIcfin2 9701 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-pow 5266 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rab 3147 df-v 3496 df-in 3943 df-ss 3952 df-pw 4541 df-uni 4839 df-po 5474 df-so 5475 df-fin2 9708 |
This theorem is referenced by: fin2i2 9740 ssfin2 9742 enfin2i 9743 fin1a2lem13 9834 |
Copyright terms: Public domain | W3C validator |