Theorem isfin1a 10333

Description: Definition of a Ia-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)

Assertion

Ref	Expression
isfin1a	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinIa ↔ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ Fin ∨ (𝐴 ∖ 𝑦) ∈ Fin)))

Distinct variable group:   𝑦,𝐴

Allowed substitution hint:   𝑉(𝑦)

Proof of Theorem isfin1a

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pweq 4613	. . 3 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
2		difeq1 4118	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∖ 𝑦) = (𝐴 ∖ 𝑦))
3	2	eleq1d 2825	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ∖ 𝑦) ∈ Fin ↔ (𝐴 ∖ 𝑦) ∈ Fin))
4	3	orbi2d 915	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑦 ∈ Fin ∨ (𝑥 ∖ 𝑦) ∈ Fin) ↔ (𝑦 ∈ Fin ∨ (𝐴 ∖ 𝑦) ∈ Fin)))
5	1, 4	raleqbidv 3345	. 2 ⊢ (𝑥 = 𝐴 → (∀𝑦 ∈ 𝒫 𝑥(𝑦 ∈ Fin ∨ (𝑥 ∖ 𝑦) ∈ Fin) ↔ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ Fin ∨ (𝐴 ∖ 𝑦) ∈ Fin)))
6		df-fin1a 10326	. 2 ⊢ FinIa = {𝑥 ∣ ∀𝑦 ∈ 𝒫 𝑥(𝑦 ∈ Fin ∨ (𝑥 ∖ 𝑦) ∈ Fin)}
7	5, 6	elab2g 3679	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinIa ↔ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ Fin ∨ (𝐴 ∖ 𝑦) ∈ Fin)))

Syntax hints:   → wi 4   ↔ wb 206   ∨ wo 847   = wceq 1539   ∈ wcel 2107  ∀wral 3060   ∖ cdif 3947  𝒫 cpw 4599  Fincfn 8986  Fin^Iacfin1a 10319

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rab 3436  df-v 3481  df-dif 3953  df-ss 3967  df-pw 4601  df-fin1a 10326

This theorem is referenced by:  fin1ai  10334  fin11a  10424  enfin1ai  10425