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Theorem fin2i 9000
 Description: Property of a II-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
Assertion
Ref Expression
fin2i (((𝐴 ∈ FinII𝐵 ⊆ 𝒫 𝐴) ∧ (𝐵 ≠ ∅ ∧ [] Or 𝐵)) → 𝐵𝐵)

Proof of Theorem fin2i
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 pwexg 4776 . . . . 5 (𝐴 ∈ FinII → 𝒫 𝐴 ∈ V)
2 elpw2g 4754 . . . . 5 (𝒫 𝐴 ∈ V → (𝐵 ∈ 𝒫 𝒫 𝐴𝐵 ⊆ 𝒫 𝐴))
31, 2syl 17 . . . 4 (𝐴 ∈ FinII → (𝐵 ∈ 𝒫 𝒫 𝐴𝐵 ⊆ 𝒫 𝐴))
43biimpar 501 . . 3 ((𝐴 ∈ FinII𝐵 ⊆ 𝒫 𝐴) → 𝐵 ∈ 𝒫 𝒫 𝐴)
5 isfin2 8999 . . . . 5 (𝐴 ∈ FinII → (𝐴 ∈ FinII ↔ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦)))
65ibi 255 . . . 4 (𝐴 ∈ FinII → ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦))
76adantr 480 . . 3 ((𝐴 ∈ FinII𝐵 ⊆ 𝒫 𝐴) → ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦))
8 neeq1 2844 . . . . . 6 (𝑦 = 𝐵 → (𝑦 ≠ ∅ ↔ 𝐵 ≠ ∅))
9 soeq2 4979 . . . . . 6 (𝑦 = 𝐵 → ( [] Or 𝑦 ↔ [] Or 𝐵))
108, 9anbi12d 743 . . . . 5 (𝑦 = 𝐵 → ((𝑦 ≠ ∅ ∧ [] Or 𝑦) ↔ (𝐵 ≠ ∅ ∧ [] Or 𝐵)))
11 unieq 4380 . . . . . 6 (𝑦 = 𝐵 𝑦 = 𝐵)
12 id 22 . . . . . 6 (𝑦 = 𝐵𝑦 = 𝐵)
1311, 12eleq12d 2682 . . . . 5 (𝑦 = 𝐵 → ( 𝑦𝑦 𝐵𝐵))
1410, 13imbi12d 333 . . . 4 (𝑦 = 𝐵 → (((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) ↔ ((𝐵 ≠ ∅ ∧ [] Or 𝐵) → 𝐵𝐵)))
1514rspcv 3278 . . 3 (𝐵 ∈ 𝒫 𝒫 𝐴 → (∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) → ((𝐵 ≠ ∅ ∧ [] Or 𝐵) → 𝐵𝐵)))
164, 7, 15sylc 63 . 2 ((𝐴 ∈ FinII𝐵 ⊆ 𝒫 𝐴) → ((𝐵 ≠ ∅ ∧ [] Or 𝐵) → 𝐵𝐵))
1716imp 444 1 (((𝐴 ∈ FinII𝐵 ⊆ 𝒫 𝐴) ∧ (𝐵 ≠ ∅ ∧ [] Or 𝐵)) → 𝐵𝐵)
 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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