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Theorem fin2i 10335
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 neeq1 3003 . . . . 5 (𝑦 = 𝐵 → (𝑦 ≠ ∅ ↔ 𝐵 ≠ ∅))
2 soeq2 5614 . . . . 5 (𝑦 = 𝐵 → ( [] Or 𝑦 ↔ [] Or 𝐵))
31, 2anbi12d 632 . . . 4 (𝑦 = 𝐵 → ((𝑦 ≠ ∅ ∧ [] Or 𝑦) ↔ (𝐵 ≠ ∅ ∧ [] Or 𝐵)))
4 unieq 4918 . . . . 5 (𝑦 = 𝐵 𝑦 = 𝐵)
5 id 22 . . . . 5 (𝑦 = 𝐵𝑦 = 𝐵)
64, 5eleq12d 2835 . . . 4 (𝑦 = 𝐵 → ( 𝑦𝑦 𝐵𝐵))
73, 6imbi12d 344 . . 3 (𝑦 = 𝐵 → (((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) ↔ ((𝐵 ≠ ∅ ∧ [] Or 𝐵) → 𝐵𝐵)))
8 isfin2 10334 . . . . 5 (𝐴 ∈ FinII → (𝐴 ∈ FinII ↔ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦)))
98ibi 267 . . . 4 (𝐴 ∈ FinII → ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦))
109adantr 480 . . 3 ((𝐴 ∈ FinII𝐵 ⊆ 𝒫 𝐴) → ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦))
11 pwexg 5378 . . . . 5 (𝐴 ∈ FinII → 𝒫 𝐴 ∈ V)
12 elpw2g 5333 . . . . 5 (𝒫 𝐴 ∈ V → (𝐵 ∈ 𝒫 𝒫 𝐴𝐵 ⊆ 𝒫 𝐴))
1311, 12syl 17 . . . 4 (𝐴 ∈ FinII → (𝐵 ∈ 𝒫 𝒫 𝐴𝐵 ⊆ 𝒫 𝐴))
1413biimpar 477 . . 3 ((𝐴 ∈ FinII𝐵 ⊆ 𝒫 𝐴) → 𝐵 ∈ 𝒫 𝒫 𝐴)
157, 10, 14rspcdva 3623 . 2 ((𝐴 ∈ FinII𝐵 ⊆ 𝒫 𝐴) → ((𝐵 ≠ ∅ ∧ [] Or 𝐵) → 𝐵𝐵))
1615imp 406 1 (((𝐴 ∈ FinII𝐵 ⊆ 𝒫 𝐴) ∧ (𝐵 ≠ ∅ ∧ [] Or 𝐵)) → 𝐵𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wne 2940  wral 3061  Vcvv 3480  wss 3951  c0 4333  𝒫 cpw 4600   cuni 4907   Or wor 5591   [] crpss 7742  FinIIcfin2 10319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-pow 5365
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-in 3958  df-ss 3968  df-pw 4602  df-uni 4908  df-po 5592  df-so 5593  df-fin2 10326
This theorem is referenced by:  fin2i2  10358  ssfin2  10360  enfin2i  10361  fin1a2lem13  10452
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