Theorem isfin1a 9326

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 4305	. . 3 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
2		difeq1 3864	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∖ 𝑦) = (𝐴 ∖ 𝑦))
3	2	eleq1d 2824	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ∖ 𝑦) ∈ Fin ↔ (𝐴 ∖ 𝑦) ∈ Fin))
4	3	orbi2d 740	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑦 ∈ Fin ∨ (𝑥 ∖ 𝑦) ∈ Fin) ↔ (𝑦 ∈ Fin ∨ (𝐴 ∖ 𝑦) ∈ Fin)))
5	1, 4	raleqbidv 3291	. 2 ⊢ (𝑥 = 𝐴 → (∀𝑦 ∈ 𝒫 𝑥(𝑦 ∈ Fin ∨ (𝑥 ∖ 𝑦) ∈ Fin) ↔ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ Fin ∨ (𝐴 ∖ 𝑦) ∈ Fin)))
6		df-fin1a 9319	. 2 ⊢ FinIa = {𝑥 ∣ ∀𝑦 ∈ 𝒫 𝑥(𝑦 ∈ Fin ∨ (𝑥 ∖ 𝑦) ∈ Fin)}
7	5, 6	elab2g 3493	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinIa ↔ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ Fin ∨ (𝐴 ∖ 𝑦) ∈ Fin)))

Syntax hints:   → wi 4   ↔ wb 196   ∨ wo 382   = wceq 1632   ∈ wcel 2139  ∀wral 3050   ∖ cdif 3712  𝒫 cpw 4302  Fincfn 8123  Fin^Iacfin1a 9312

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rab 3059  df-v 3342  df-dif 3718  df-in 3722  df-ss 3729  df-pw 4304  df-fin1a 9319

This theorem is referenced by:  fin1ai  9327  fin11a  9417  enfin1ai  9418