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Theorem isfin2 9710
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 variables 𝑥 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pweq 4538 . . . 4 (𝑧 = 𝑥 → 𝒫 𝑧 = 𝒫 𝑥)
21pweqd 4541 . . 3 (𝑧 = 𝑥 → 𝒫 𝒫 𝑧 = 𝒫 𝒫 𝑥)
32raleqdv 3403 . 2 (𝑧 = 𝑥 → (∀𝑦 ∈ 𝒫 𝒫 𝑧((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) ↔ ∀𝑦 ∈ 𝒫 𝒫 𝑥((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦)))
4 pweq 4538 . . . 4 (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
54pweqd 4541 . . 3 (𝑥 = 𝐴 → 𝒫 𝒫 𝑥 = 𝒫 𝒫 𝐴)
65raleqdv 3403 . 2 (𝑥 = 𝐴 → (∀𝑦 ∈ 𝒫 𝒫 𝑥((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦) ↔ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦)))
7 df-fin2 9702 . 2 FinII = {𝑧 ∣ ∀𝑦 ∈ 𝒫 𝒫 𝑧((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦)}
83, 6, 7elab2gw 3651 1 (𝐴𝑉 → (𝐴 ∈ FinII ↔ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  wne 3014  wral 3133  c0 4276  𝒫 cpw 4522   cuni 4825   Or wor 5461   [] crpss 7439  FinIIcfin2 9695
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-ral 3138  df-v 3482  df-in 3926  df-ss 3936  df-pw 4524  df-fin2 9702
This theorem is referenced by:  fin2i  9711  isfin2-2  9735  ssfin2  9736  enfin2i  9737  fin12  9829  fin1a2s  9830
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