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Theorem isfin2 9154
Description: Definition of a II-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
Assertion
Ref Expression
isfin2 (𝐴𝑉 → (𝐴 ∈ FinII ↔ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦)))
Distinct variable group:   𝑦,𝐴
Allowed substitution hint:   𝑉(𝑦)

Proof of Theorem isfin2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 pweq 4194 . . . 4 (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
21pweqd 4196 . . 3 (𝑥 = 𝐴 → 𝒫 𝒫 𝑥 = 𝒫 𝒫 𝐴)
32raleqdv 3174 . 2 (𝑥 = 𝐴 → (∀𝑦 ∈ 𝒫 𝒫 𝑥((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) ↔ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦)))
4 df-fin2 9146 . 2 FinII = {𝑥 ∣ ∀𝑦 ∈ 𝒫 𝒫 𝑥((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦)}
53, 4elab2g 3385 1 (𝐴𝑉 → (𝐴 ∈ FinII ↔ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wne 2823  wral 2941  c0 3948  𝒫 cpw 4191   cuni 4468   Or wor 5063   [] crpss 6978  FinIIcfin2 9139
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-v 3233  df-in 3614  df-ss 3621  df-pw 4193  df-fin2 9146
This theorem is referenced by:  fin2i  9155  isfin2-2  9179  ssfin2  9180  enfin2i  9181  fin12  9273  fin1a2s  9274
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