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Theorem fin2i 9061
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 4810 . . . . 5 (𝐴 ∈ FinII → 𝒫 𝐴 ∈ V)
2 elpw2g 4787 . . . . 5 (𝒫 𝐴 ∈ V → (𝐵 ∈ 𝒫 𝒫 𝐴𝐵 ⊆ 𝒫 𝐴))
31, 2syl 17 . . . 4 (𝐴 ∈ FinII → (𝐵 ∈ 𝒫 𝒫 𝐴𝐵 ⊆ 𝒫 𝐴))
43biimpar 502 . . 3 ((𝐴 ∈ FinII𝐵 ⊆ 𝒫 𝐴) → 𝐵 ∈ 𝒫 𝒫 𝐴)
5 isfin2 9060 . . . . 5 (𝐴 ∈ FinII → (𝐴 ∈ FinII ↔ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦)))
65ibi 256 . . . 4 (𝐴 ∈ FinII → ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦))
76adantr 481 . . 3 ((𝐴 ∈ FinII𝐵 ⊆ 𝒫 𝐴) → ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦))
8 neeq1 2852 . . . . . 6 (𝑦 = 𝐵 → (𝑦 ≠ ∅ ↔ 𝐵 ≠ ∅))
9 soeq2 5015 . . . . . 6 (𝑦 = 𝐵 → ( [] Or 𝑦 ↔ [] Or 𝐵))
108, 9anbi12d 746 . . . . 5 (𝑦 = 𝐵 → ((𝑦 ≠ ∅ ∧ [] Or 𝑦) ↔ (𝐵 ≠ ∅ ∧ [] Or 𝐵)))
11 unieq 4410 . . . . . 6 (𝑦 = 𝐵 𝑦 = 𝐵)
12 id 22 . . . . . 6 (𝑦 = 𝐵𝑦 = 𝐵)
1311, 12eleq12d 2692 . . . . 5 (𝑦 = 𝐵 → ( 𝑦𝑦 𝐵𝐵))
1410, 13imbi12d 334 . . . 4 (𝑦 = 𝐵 → (((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) ↔ ((𝐵 ≠ ∅ ∧ [] Or 𝐵) → 𝐵𝐵)))
1514rspcv 3291 . . 3 (𝐵 ∈ 𝒫 𝒫 𝐴 → (∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) → ((𝐵 ≠ ∅ ∧ [] Or 𝐵) → 𝐵𝐵)))
164, 7, 15sylc 65 . 2 ((𝐴 ∈ FinII𝐵 ⊆ 𝒫 𝐴) → ((𝐵 ≠ ∅ ∧ [] Or 𝐵) → 𝐵𝐵))
1716imp 445 1 (((𝐴 ∈ FinII𝐵 ⊆ 𝒫 𝐴) ∧ (𝐵 ≠ ∅ ∧ [] Or 𝐵)) → 𝐵𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wne 2790  wral 2907  Vcvv 3186  wss 3555  c0 3891  𝒫 cpw 4130   cuni 4402   Or wor 4994   [] crpss 6889  FinIIcfin2 9045
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-pow 4803
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-v 3188  df-in 3562  df-ss 3569  df-pw 4132  df-uni 4403  df-po 4995  df-so 4996  df-fin2 9052
This theorem is referenced by:  fin2i2  9084  ssfin2  9086  enfin2i  9087  fin1a2lem13  9178
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