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Theorem fin2i 9319
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 3005 . . . . 5 (𝑦 = 𝐵 → (𝑦 ≠ ∅ ↔ 𝐵 ≠ ∅))
2 soeq2 5190 . . . . 5 (𝑦 = 𝐵 → ( [] Or 𝑦 ↔ [] Or 𝐵))
31, 2anbi12d 616 . . . 4 (𝑦 = 𝐵 → ((𝑦 ≠ ∅ ∧ [] Or 𝑦) ↔ (𝐵 ≠ ∅ ∧ [] Or 𝐵)))
4 unieq 4582 . . . . 5 (𝑦 = 𝐵 𝑦 = 𝐵)
5 id 22 . . . . 5 (𝑦 = 𝐵𝑦 = 𝐵)
64, 5eleq12d 2844 . . . 4 (𝑦 = 𝐵 → ( 𝑦𝑦 𝐵𝐵))
73, 6imbi12d 333 . . 3 (𝑦 = 𝐵 → (((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) ↔ ((𝐵 ≠ ∅ ∧ [] Or 𝐵) → 𝐵𝐵)))
8 isfin2 9318 . . . . 5 (𝐴 ∈ FinII → (𝐴 ∈ FinII ↔ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦)))
98ibi 256 . . . 4 (𝐴 ∈ FinII → ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦))
109adantr 466 . . 3 ((𝐴 ∈ FinII𝐵 ⊆ 𝒫 𝐴) → ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦))
11 pwexg 4980 . . . . 5 (𝐴 ∈ FinII → 𝒫 𝐴 ∈ V)
12 elpw2g 4958 . . . . 5 (𝒫 𝐴 ∈ V → (𝐵 ∈ 𝒫 𝒫 𝐴𝐵 ⊆ 𝒫 𝐴))
1311, 12syl 17 . . . 4 (𝐴 ∈ FinII → (𝐵 ∈ 𝒫 𝒫 𝐴𝐵 ⊆ 𝒫 𝐴))
1413biimpar 463 . . 3 ((𝐴 ∈ FinII𝐵 ⊆ 𝒫 𝐴) → 𝐵 ∈ 𝒫 𝒫 𝐴)
157, 10, 14rspcdva 3466 . 2 ((𝐴 ∈ FinII𝐵 ⊆ 𝒫 𝐴) → ((𝐵 ≠ ∅ ∧ [] Or 𝐵) → 𝐵𝐵))
1615imp 393 1 (((𝐴 ∈ FinII𝐵 ⊆ 𝒫 𝐴) ∧ (𝐵 ≠ ∅ ∧ [] Or 𝐵)) → 𝐵𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wne 2943  wral 3061  Vcvv 3351  wss 3723  c0 4063  𝒫 cpw 4297   cuni 4574   Or wor 5169   [] crpss 7083  FinIIcfin2 9303
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-pow 4974
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-v 3353  df-in 3730  df-ss 3737  df-pw 4299  df-uni 4575  df-po 5170  df-so 5171  df-fin2 9310
This theorem is referenced by:  fin2i2  9342  ssfin2  9344  enfin2i  9345  fin1a2lem13  9436
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