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Theorem isfin1a 9706
 Description: Definition of a Ia-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
Assertion
Ref Expression
isfin1a (𝐴𝑉 → (𝐴 ∈ FinIa ↔ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ Fin ∨ (𝐴𝑦) ∈ Fin)))
Distinct variable group:   𝑦,𝐴
Allowed substitution hint:   𝑉(𝑦)

Proof of Theorem isfin1a
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 pweq 4544 . . 3 (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
2 difeq1 4095 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑦) = (𝐴𝑦))
32eleq1d 2901 . . . 4 (𝑥 = 𝐴 → ((𝑥𝑦) ∈ Fin ↔ (𝐴𝑦) ∈ Fin))
43orbi2d 911 . . 3 (𝑥 = 𝐴 → ((𝑦 ∈ Fin ∨ (𝑥𝑦) ∈ Fin) ↔ (𝑦 ∈ Fin ∨ (𝐴𝑦) ∈ Fin)))
51, 4raleqbidv 3406 . 2 (𝑥 = 𝐴 → (∀𝑦 ∈ 𝒫 𝑥(𝑦 ∈ Fin ∨ (𝑥𝑦) ∈ Fin) ↔ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ Fin ∨ (𝐴𝑦) ∈ Fin)))
6 df-fin1a 9699 . 2 FinIa = {𝑥 ∣ ∀𝑦 ∈ 𝒫 𝑥(𝑦 ∈ Fin ∨ (𝑥𝑦) ∈ Fin)}
75, 6elab2g 3672 1 (𝐴𝑉 → (𝐴 ∈ FinIa ↔ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ Fin ∨ (𝐴𝑦) ∈ Fin)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   ∨ wo 843   = wceq 1530   ∈ wcel 2107  ∀wral 3142   ∖ cdif 3936  𝒫 cpw 4541  Fincfn 8501  FinIacfin1a 9692 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ral 3147  df-rab 3151  df-dif 3942  df-in 3946  df-ss 3955  df-pw 4543  df-fin1a 9699 This theorem is referenced by:  fin1ai  9707  fin11a  9797  enfin1ai  9798
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