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Mirrors > Home > MPE Home > Th. List > ssfin2 | Structured version Visualization version GIF version |
Description: A subset of a II-finite set is II-finite. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Revised by Mario Carneiro, 16-May-2015.) |
Ref | Expression |
---|---|
ssfin2 | ⊢ ((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ FinII) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 765 | . . . 4 ⊢ (((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ 𝒫 𝒫 𝐵) → 𝐴 ∈ FinII) | |
2 | elpwi 4565 | . . . . . 6 ⊢ (𝑥 ∈ 𝒫 𝒫 𝐵 → 𝑥 ⊆ 𝒫 𝐵) | |
3 | 2 | adantl 482 | . . . . 5 ⊢ (((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ 𝒫 𝒫 𝐵) → 𝑥 ⊆ 𝒫 𝐵) |
4 | simplr 767 | . . . . . 6 ⊢ (((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ 𝒫 𝒫 𝐵) → 𝐵 ⊆ 𝐴) | |
5 | 4 | sspwd 4571 | . . . . 5 ⊢ (((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ 𝒫 𝒫 𝐵) → 𝒫 𝐵 ⊆ 𝒫 𝐴) |
6 | 3, 5 | sstrd 3952 | . . . 4 ⊢ (((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ 𝒫 𝒫 𝐵) → 𝑥 ⊆ 𝒫 𝐴) |
7 | fin2i 10227 | . . . . 5 ⊢ (((𝐴 ∈ FinII ∧ 𝑥 ⊆ 𝒫 𝐴) ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥)) → ∪ 𝑥 ∈ 𝑥) | |
8 | 7 | ex 413 | . . . 4 ⊢ ((𝐴 ∈ FinII ∧ 𝑥 ⊆ 𝒫 𝐴) → ((𝑥 ≠ ∅ ∧ [⊊] Or 𝑥) → ∪ 𝑥 ∈ 𝑥)) |
9 | 1, 6, 8 | syl2anc 584 | . . 3 ⊢ (((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ 𝒫 𝒫 𝐵) → ((𝑥 ≠ ∅ ∧ [⊊] Or 𝑥) → ∪ 𝑥 ∈ 𝑥)) |
10 | 9 | ralrimiva 3141 | . 2 ⊢ ((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝐴) → ∀𝑥 ∈ 𝒫 𝒫 𝐵((𝑥 ≠ ∅ ∧ [⊊] Or 𝑥) → ∪ 𝑥 ∈ 𝑥)) |
11 | ssexg 5278 | . . . 4 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ∈ FinII) → 𝐵 ∈ V) | |
12 | 11 | ancoms 459 | . . 3 ⊢ ((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ V) |
13 | isfin2 10226 | . . 3 ⊢ (𝐵 ∈ V → (𝐵 ∈ FinII ↔ ∀𝑥 ∈ 𝒫 𝒫 𝐵((𝑥 ≠ ∅ ∧ [⊊] Or 𝑥) → ∪ 𝑥 ∈ 𝑥))) | |
14 | 12, 13 | syl 17 | . 2 ⊢ ((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝐴) → (𝐵 ∈ FinII ↔ ∀𝑥 ∈ 𝒫 𝒫 𝐵((𝑥 ≠ ∅ ∧ [⊊] Or 𝑥) → ∪ 𝑥 ∈ 𝑥))) |
15 | 10, 14 | mpbird 256 | 1 ⊢ ((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ FinII) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∈ wcel 2106 ≠ wne 2941 ∀wral 3062 Vcvv 3443 ⊆ wss 3908 ∅c0 4280 𝒫 cpw 4558 ∪ cuni 4863 Or wor 5542 [⊊] crpss 7655 FinIIcfin2 10211 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 ax-sep 5254 ax-pow 5318 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-ne 2942 df-ral 3063 df-rab 3406 df-v 3445 df-in 3915 df-ss 3925 df-pw 4560 df-uni 4864 df-po 5543 df-so 5544 df-fin2 10218 |
This theorem is referenced by: (None) |
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