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Theorem ssfin2 9139
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 790 . . . 4 (((𝐴 ∈ FinII𝐵𝐴) ∧ 𝑥 ∈ 𝒫 𝒫 𝐵) → 𝐴 ∈ FinII)
2 elpwi 4166 . . . . . 6 (𝑥 ∈ 𝒫 𝒫 𝐵𝑥 ⊆ 𝒫 𝐵)
32adantl 482 . . . . 5 (((𝐴 ∈ FinII𝐵𝐴) ∧ 𝑥 ∈ 𝒫 𝒫 𝐵) → 𝑥 ⊆ 𝒫 𝐵)
4 simplr 792 . . . . . 6 (((𝐴 ∈ FinII𝐵𝐴) ∧ 𝑥 ∈ 𝒫 𝒫 𝐵) → 𝐵𝐴)
5 sspwb 4915 . . . . . 6 (𝐵𝐴 ↔ 𝒫 𝐵 ⊆ 𝒫 𝐴)
64, 5sylib 208 . . . . 5 (((𝐴 ∈ FinII𝐵𝐴) ∧ 𝑥 ∈ 𝒫 𝒫 𝐵) → 𝒫 𝐵 ⊆ 𝒫 𝐴)
73, 6sstrd 3611 . . . 4 (((𝐴 ∈ FinII𝐵𝐴) ∧ 𝑥 ∈ 𝒫 𝒫 𝐵) → 𝑥 ⊆ 𝒫 𝐴)
8 fin2i 9114 . . . . 5 (((𝐴 ∈ FinII𝑥 ⊆ 𝒫 𝐴) ∧ (𝑥 ≠ ∅ ∧ [] Or 𝑥)) → 𝑥𝑥)
98ex 450 . . . 4 ((𝐴 ∈ FinII𝑥 ⊆ 𝒫 𝐴) → ((𝑥 ≠ ∅ ∧ [] Or 𝑥) → 𝑥𝑥))
101, 7, 9syl2anc 693 . . 3 (((𝐴 ∈ FinII𝐵𝐴) ∧ 𝑥 ∈ 𝒫 𝒫 𝐵) → ((𝑥 ≠ ∅ ∧ [] Or 𝑥) → 𝑥𝑥))
1110ralrimiva 2965 . 2 ((𝐴 ∈ FinII𝐵𝐴) → ∀𝑥 ∈ 𝒫 𝒫 𝐵((𝑥 ≠ ∅ ∧ [] Or 𝑥) → 𝑥𝑥))
12 ssexg 4802 . . . 4 ((𝐵𝐴𝐴 ∈ FinII) → 𝐵 ∈ V)
1312ancoms 469 . . 3 ((𝐴 ∈ FinII𝐵𝐴) → 𝐵 ∈ V)
14 isfin2 9113 . . 3 (𝐵 ∈ V → (𝐵 ∈ FinII ↔ ∀𝑥 ∈ 𝒫 𝒫 𝐵((𝑥 ≠ ∅ ∧ [] Or 𝑥) → 𝑥𝑥)))
1513, 14syl 17 . 2 ((𝐴 ∈ FinII𝐵𝐴) → (𝐵 ∈ FinII ↔ ∀𝑥 ∈ 𝒫 𝒫 𝐵((𝑥 ≠ ∅ ∧ [] Or 𝑥) → 𝑥𝑥)))
1611, 15mpbird 247 1 ((𝐴 ∈ FinII𝐵𝐴) → 𝐵 ∈ FinII)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wcel 1989  wne 2793  wral 2911  Vcvv 3198  wss 3572  c0 3913  𝒫 cpw 4156   cuni 4434   Or wor 5032   [] crpss 6933  FinIIcfin2 9098
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-pw 4158  df-sn 4176  df-pr 4178  df-uni 4435  df-po 5033  df-so 5034  df-fin2 9105
This theorem is referenced by: (None)
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