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Theorem ssfin2 9354
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 807 . . . 4 (((𝐴 ∈ FinII𝐵𝐴) ∧ 𝑥 ∈ 𝒫 𝒫 𝐵) → 𝐴 ∈ FinII)
2 elpwi 4312 . . . . . 6 (𝑥 ∈ 𝒫 𝒫 𝐵𝑥 ⊆ 𝒫 𝐵)
32adantl 473 . . . . 5 (((𝐴 ∈ FinII𝐵𝐴) ∧ 𝑥 ∈ 𝒫 𝒫 𝐵) → 𝑥 ⊆ 𝒫 𝐵)
4 simplr 809 . . . . . 6 (((𝐴 ∈ FinII𝐵𝐴) ∧ 𝑥 ∈ 𝒫 𝒫 𝐵) → 𝐵𝐴)
5 sspwb 5066 . . . . . 6 (𝐵𝐴 ↔ 𝒫 𝐵 ⊆ 𝒫 𝐴)
64, 5sylib 208 . . . . 5 (((𝐴 ∈ FinII𝐵𝐴) ∧ 𝑥 ∈ 𝒫 𝒫 𝐵) → 𝒫 𝐵 ⊆ 𝒫 𝐴)
73, 6sstrd 3754 . . . 4 (((𝐴 ∈ FinII𝐵𝐴) ∧ 𝑥 ∈ 𝒫 𝒫 𝐵) → 𝑥 ⊆ 𝒫 𝐴)
8 fin2i 9329 . . . . 5 (((𝐴 ∈ FinII𝑥 ⊆ 𝒫 𝐴) ∧ (𝑥 ≠ ∅ ∧ [] Or 𝑥)) → 𝑥𝑥)
98ex 449 . . . 4 ((𝐴 ∈ FinII𝑥 ⊆ 𝒫 𝐴) → ((𝑥 ≠ ∅ ∧ [] Or 𝑥) → 𝑥𝑥))
101, 7, 9syl2anc 696 . . 3 (((𝐴 ∈ FinII𝐵𝐴) ∧ 𝑥 ∈ 𝒫 𝒫 𝐵) → ((𝑥 ≠ ∅ ∧ [] Or 𝑥) → 𝑥𝑥))
1110ralrimiva 3104 . 2 ((𝐴 ∈ FinII𝐵𝐴) → ∀𝑥 ∈ 𝒫 𝒫 𝐵((𝑥 ≠ ∅ ∧ [] Or 𝑥) → 𝑥𝑥))
12 ssexg 4956 . . . 4 ((𝐵𝐴𝐴 ∈ FinII) → 𝐵 ∈ V)
1312ancoms 468 . . 3 ((𝐴 ∈ FinII𝐵𝐴) → 𝐵 ∈ V)
14 isfin2 9328 . . 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 383  wcel 2139  wne 2932  wral 3050  Vcvv 3340  wss 3715  c0 4058  𝒫 cpw 4302   cuni 4588   Or wor 5186   [] crpss 7102  FinIIcfin2 9313
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-pw 4304  df-sn 4322  df-pr 4324  df-uni 4589  df-po 5187  df-so 5188  df-fin2 9320
This theorem is referenced by: (None)
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