Theorem fin1ai 9305

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

Assertion

Ref	Expression
fin1ai	⊢ ((𝐴 ∈ FinIa ∧ 𝑋 ⊆ 𝐴) → (𝑋 ∈ Fin ∨ (𝐴 ∖ 𝑋) ∈ Fin))

Proof of Theorem fin1ai

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2825	. . 3 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ Fin ↔ 𝑋 ∈ Fin))
2		difeq2 3863	. . . 4 ⊢ (𝑥 = 𝑋 → (𝐴 ∖ 𝑥) = (𝐴 ∖ 𝑋))
3	2	eleq1d 2822	. . 3 ⊢ (𝑥 = 𝑋 → ((𝐴 ∖ 𝑥) ∈ Fin ↔ (𝐴 ∖ 𝑋) ∈ Fin))
4	1, 3	orbi12d 748	. 2 ⊢ (𝑥 = 𝑋 → ((𝑥 ∈ Fin ∨ (𝐴 ∖ 𝑥) ∈ Fin) ↔ (𝑋 ∈ Fin ∨ (𝐴 ∖ 𝑋) ∈ Fin)))
5		isfin1a 9304	. . . 4 ⊢ (𝐴 ∈ FinIa → (𝐴 ∈ FinIa ↔ ∀𝑥 ∈ 𝒫 𝐴(𝑥 ∈ Fin ∨ (𝐴 ∖ 𝑥) ∈ Fin)))
6	5	ibi 256	. . 3 ⊢ (𝐴 ∈ FinIa → ∀𝑥 ∈ 𝒫 𝐴(𝑥 ∈ Fin ∨ (𝐴 ∖ 𝑥) ∈ Fin))
7	6	adantr 472	. 2 ⊢ ((𝐴 ∈ FinIa ∧ 𝑋 ⊆ 𝐴) → ∀𝑥 ∈ 𝒫 𝐴(𝑥 ∈ Fin ∨ (𝐴 ∖ 𝑥) ∈ Fin))
8		elpw2g 4974	. . 3 ⊢ (𝐴 ∈ FinIa → (𝑋 ∈ 𝒫 𝐴 ↔ 𝑋 ⊆ 𝐴))
9	8	biimpar 503	. 2 ⊢ ((𝐴 ∈ FinIa ∧ 𝑋 ⊆ 𝐴) → 𝑋 ∈ 𝒫 𝐴)
10	4, 7, 9	rspcdva 3453	1 ⊢ ((𝐴 ∈ FinIa ∧ 𝑋 ⊆ 𝐴) → (𝑋 ∈ Fin ∨ (𝐴 ∖ 𝑋) ∈ Fin))

Syntax hints:   → wi 4   ∨ wo 382   ∧ wa 383   = wceq 1630   ∈ wcel 2137  ∀wral 3048   ∖ cdif 3710   ⊆ wss 3713  𝒫 cpw 4300  Fincfn 8119  Fin^Iacfin1a 9290

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ral 3053  df-rab 3057  df-v 3340  df-dif 3716  df-in 3720  df-ss 3727  df-pw 4302  df-fin1a 9297

This theorem is referenced by:  enfin1ai  9396  fin1a2  9427  fin1aufil  21935