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Theorem ssfin2 10252
Description: A subset of a II-finite set is II-finite. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Revised by Mario Carneiro, 16-May-2015.)
Assertion
Ref Expression
ssfin2 ((𝐴 ∈ FinII𝐵𝐴) → 𝐵 ∈ FinII)

Proof of Theorem ssfin2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 simpll 765 . . . 4 (((𝐴 ∈ FinII𝐵𝐴) ∧ 𝑥 ∈ 𝒫 𝒫 𝐵) → 𝐴 ∈ FinII)
2 elpwi 4565 . . . . . 6 (𝑥 ∈ 𝒫 𝒫 𝐵𝑥 ⊆ 𝒫 𝐵)
32adantl 482 . . . . 5 (((𝐴 ∈ FinII𝐵𝐴) ∧ 𝑥 ∈ 𝒫 𝒫 𝐵) → 𝑥 ⊆ 𝒫 𝐵)
4 simplr 767 . . . . . 6 (((𝐴 ∈ FinII𝐵𝐴) ∧ 𝑥 ∈ 𝒫 𝒫 𝐵) → 𝐵𝐴)
54sspwd 4571 . . . . 5 (((𝐴 ∈ FinII𝐵𝐴) ∧ 𝑥 ∈ 𝒫 𝒫 𝐵) → 𝒫 𝐵 ⊆ 𝒫 𝐴)
63, 5sstrd 3952 . . . 4 (((𝐴 ∈ FinII𝐵𝐴) ∧ 𝑥 ∈ 𝒫 𝒫 𝐵) → 𝑥 ⊆ 𝒫 𝐴)
7 fin2i 10227 . . . . 5 (((𝐴 ∈ FinII𝑥 ⊆ 𝒫 𝐴) ∧ (𝑥 ≠ ∅ ∧ [] Or 𝑥)) → 𝑥𝑥)
87ex 413 . . . 4 ((𝐴 ∈ FinII𝑥 ⊆ 𝒫 𝐴) → ((𝑥 ≠ ∅ ∧ [] Or 𝑥) → 𝑥𝑥))
91, 6, 8syl2anc 584 . . 3 (((𝐴 ∈ FinII𝐵𝐴) ∧ 𝑥 ∈ 𝒫 𝒫 𝐵) → ((𝑥 ≠ ∅ ∧ [] Or 𝑥) → 𝑥𝑥))
109ralrimiva 3141 . 2 ((𝐴 ∈ FinII𝐵𝐴) → ∀𝑥 ∈ 𝒫 𝒫 𝐵((𝑥 ≠ ∅ ∧ [] Or 𝑥) → 𝑥𝑥))
11 ssexg 5278 . . . 4 ((𝐵𝐴𝐴 ∈ FinII) → 𝐵 ∈ V)
1211ancoms 459 . . 3 ((𝐴 ∈ FinII𝐵𝐴) → 𝐵 ∈ V)
13 isfin2 10226 . . 3 (𝐵 ∈ V → (𝐵 ∈ FinII ↔ ∀𝑥 ∈ 𝒫 𝒫 𝐵((𝑥 ≠ ∅ ∧ [] Or 𝑥) → 𝑥𝑥)))
1412, 13syl 17 . 2 ((𝐴 ∈ FinII𝐵𝐴) → (𝐵 ∈ FinII ↔ ∀𝑥 ∈ 𝒫 𝒫 𝐵((𝑥 ≠ ∅ ∧ [] Or 𝑥) → 𝑥𝑥)))
1510, 14mpbird 256 1 ((𝐴 ∈ FinII𝐵𝐴) → 𝐵 ∈ FinII)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wcel 2106  wne 2941  wral 3062  Vcvv 3443  wss 3908  c0 4280  𝒫 cpw 4558   cuni 4863   Or wor 5542   [] crpss 7655  FinIIcfin2 10211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-sep 5254  ax-pow 5318
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2942  df-ral 3063  df-rab 3406  df-v 3445  df-in 3915  df-ss 3925  df-pw 4560  df-uni 4864  df-po 5543  df-so 5544  df-fin2 10218
This theorem is referenced by: (None)
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