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Theorem isfin2 9101
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 4152 . . . 4 (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
21pweqd 4154 . . 3 (𝑥 = 𝐴 → 𝒫 𝒫 𝑥 = 𝒫 𝒫 𝐴)
32raleqdv 3139 . 2 (𝑥 = 𝐴 → (∀𝑦 ∈ 𝒫 𝒫 𝑥((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) ↔ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦)))
4 df-fin2 9093 . 2 FinII = {𝑥 ∣ ∀𝑦 ∈ 𝒫 𝒫 𝑥((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦)}
53, 4elab2g 3347 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 384   = wceq 1481  wcel 1988  wne 2791  wral 2909  c0 3907  𝒫 cpw 4149   cuni 4427   Or wor 5024   [] crpss 6921  FinIIcfin2 9086
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-v 3197  df-in 3574  df-ss 3581  df-pw 4151  df-fin2 9093
This theorem is referenced by:  fin2i  9102  isfin2-2  9126  ssfin2  9127  enfin2i  9128  fin12  9220  fin1a2s  9221
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