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Theorem ssfin2 9734
 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 766 . . . 4 (((𝐴 ∈ FinII𝐵𝐴) ∧ 𝑥 ∈ 𝒫 𝒫 𝐵) → 𝐴 ∈ FinII)
2 elpwi 4506 . . . . . 6 (𝑥 ∈ 𝒫 𝒫 𝐵𝑥 ⊆ 𝒫 𝐵)
32adantl 485 . . . . 5 (((𝐴 ∈ FinII𝐵𝐴) ∧ 𝑥 ∈ 𝒫 𝒫 𝐵) → 𝑥 ⊆ 𝒫 𝐵)
4 simplr 768 . . . . . 6 (((𝐴 ∈ FinII𝐵𝐴) ∧ 𝑥 ∈ 𝒫 𝒫 𝐵) → 𝐵𝐴)
54sspwd 4512 . . . . 5 (((𝐴 ∈ FinII𝐵𝐴) ∧ 𝑥 ∈ 𝒫 𝒫 𝐵) → 𝒫 𝐵 ⊆ 𝒫 𝐴)
63, 5sstrd 3925 . . . 4 (((𝐴 ∈ FinII𝐵𝐴) ∧ 𝑥 ∈ 𝒫 𝒫 𝐵) → 𝑥 ⊆ 𝒫 𝐴)
7 fin2i 9709 . . . . 5 (((𝐴 ∈ FinII𝑥 ⊆ 𝒫 𝐴) ∧ (𝑥 ≠ ∅ ∧ [] Or 𝑥)) → 𝑥𝑥)
87ex 416 . . . 4 ((𝐴 ∈ FinII𝑥 ⊆ 𝒫 𝐴) → ((𝑥 ≠ ∅ ∧ [] Or 𝑥) → 𝑥𝑥))
91, 6, 8syl2anc 587 . . 3 (((𝐴 ∈ FinII𝐵𝐴) ∧ 𝑥 ∈ 𝒫 𝒫 𝐵) → ((𝑥 ≠ ∅ ∧ [] Or 𝑥) → 𝑥𝑥))
109ralrimiva 3149 . 2 ((𝐴 ∈ FinII𝐵𝐴) → ∀𝑥 ∈ 𝒫 𝒫 𝐵((𝑥 ≠ ∅ ∧ [] Or 𝑥) → 𝑥𝑥))
11 ssexg 5192 . . . 4 ((𝐵𝐴𝐴 ∈ FinII) → 𝐵 ∈ V)
1211ancoms 462 . . 3 ((𝐴 ∈ FinII𝐵𝐴) → 𝐵 ∈ V)
13 isfin2 9708 . . 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 260 1 ((𝐴 ∈ FinII𝐵𝐴) → 𝐵 ∈ FinII)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  Vcvv 3441   ⊆ wss 3881  ∅c0 4243  𝒫 cpw 4497  ∪ cuni 4801   Or wor 5438   [⊊] crpss 7431  FinIIcfin2 9693 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-pow 5232 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ne 2988  df-ral 3111  df-rab 3115  df-v 3443  df-in 3888  df-ss 3898  df-pw 4499  df-uni 4802  df-po 5439  df-so 5440  df-fin2 9700 This theorem is referenced by: (None)
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