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Theorem fin2i 9706
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 3083 . . . . 5 (𝑦 = 𝐵 → (𝑦 ≠ ∅ ↔ 𝐵 ≠ ∅))
2 soeq2 5494 . . . . 5 (𝑦 = 𝐵 → ( [] Or 𝑦 ↔ [] Or 𝐵))
31, 2anbi12d 630 . . . 4 (𝑦 = 𝐵 → ((𝑦 ≠ ∅ ∧ [] Or 𝑦) ↔ (𝐵 ≠ ∅ ∧ [] Or 𝐵)))
4 unieq 4845 . . . . 5 (𝑦 = 𝐵 𝑦 = 𝐵)
5 id 22 . . . . 5 (𝑦 = 𝐵𝑦 = 𝐵)
64, 5eleq12d 2912 . . . 4 (𝑦 = 𝐵 → ( 𝑦𝑦 𝐵𝐵))
73, 6imbi12d 346 . . 3 (𝑦 = 𝐵 → (((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) ↔ ((𝐵 ≠ ∅ ∧ [] Or 𝐵) → 𝐵𝐵)))
8 isfin2 9705 . . . . 5 (𝐴 ∈ FinII → (𝐴 ∈ FinII ↔ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦)))
98ibi 268 . . . 4 (𝐴 ∈ FinII → ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦))
109adantr 481 . . 3 ((𝐴 ∈ FinII𝐵 ⊆ 𝒫 𝐴) → ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦))
11 pwexg 5276 . . . . 5 (𝐴 ∈ FinII → 𝒫 𝐴 ∈ V)
12 elpw2g 5244 . . . . 5 (𝒫 𝐴 ∈ V → (𝐵 ∈ 𝒫 𝒫 𝐴𝐵 ⊆ 𝒫 𝐴))
1311, 12syl 17 . . . 4 (𝐴 ∈ FinII → (𝐵 ∈ 𝒫 𝒫 𝐴𝐵 ⊆ 𝒫 𝐴))
1413biimpar 478 . . 3 ((𝐴 ∈ FinII𝐵 ⊆ 𝒫 𝐴) → 𝐵 ∈ 𝒫 𝒫 𝐴)
157, 10, 14rspcdva 3629 . 2 ((𝐴 ∈ FinII𝐵 ⊆ 𝒫 𝐴) → ((𝐵 ≠ ∅ ∧ [] Or 𝐵) → 𝐵𝐵))
1615imp 407 1 (((𝐴 ∈ FinII𝐵 ⊆ 𝒫 𝐴) ∧ (𝐵 ≠ ∅ ∧ [] Or 𝐵)) → 𝐵𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wcel 2107  wne 3021  wral 3143  Vcvv 3500  wss 3940  c0 4295  𝒫 cpw 4542   cuni 4837   Or wor 5472   [] crpss 7438  FinIIcfin2 9690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-pow 5263
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-in 3947  df-ss 3956  df-pw 4544  df-uni 4838  df-po 5473  df-so 5474  df-fin2 9697
This theorem is referenced by:  fin2i2  9729  ssfin2  9731  enfin2i  9732  fin1a2lem13  9823
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