Theorem fin1ai 10292

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 2819	. . 3 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ Fin ↔ 𝑋 ∈ Fin))
2		difeq2 4117	. . . 4 ⊢ (𝑥 = 𝑋 → (𝐴 ∖ 𝑥) = (𝐴 ∖ 𝑋))
3	2	eleq1d 2816	. . 3 ⊢ (𝑥 = 𝑋 → ((𝐴 ∖ 𝑥) ∈ Fin ↔ (𝐴 ∖ 𝑋) ∈ Fin))
4	1, 3	orbi12d 915	. 2 ⊢ (𝑥 = 𝑋 → ((𝑥 ∈ Fin ∨ (𝐴 ∖ 𝑥) ∈ Fin) ↔ (𝑋 ∈ Fin ∨ (𝐴 ∖ 𝑋) ∈ Fin)))
5		isfin1a 10291	. . . 4 ⊢ (𝐴 ∈ FinIa → (𝐴 ∈ FinIa ↔ ∀𝑥 ∈ 𝒫 𝐴(𝑥 ∈ Fin ∨ (𝐴 ∖ 𝑥) ∈ Fin)))
6	5	ibi 266	. . 3 ⊢ (𝐴 ∈ FinIa → ∀𝑥 ∈ 𝒫 𝐴(𝑥 ∈ Fin ∨ (𝐴 ∖ 𝑥) ∈ Fin))
7	6	adantr 479	. 2 ⊢ ((𝐴 ∈ FinIa ∧ 𝑋 ⊆ 𝐴) → ∀𝑥 ∈ 𝒫 𝐴(𝑥 ∈ Fin ∨ (𝐴 ∖ 𝑥) ∈ Fin))
8		elpw2g 5345	. . 3 ⊢ (𝐴 ∈ FinIa → (𝑋 ∈ 𝒫 𝐴 ↔ 𝑋 ⊆ 𝐴))
9	8	biimpar 476	. 2 ⊢ ((𝐴 ∈ FinIa ∧ 𝑋 ⊆ 𝐴) → 𝑋 ∈ 𝒫 𝐴)
10	4, 7, 9	rspcdva 3614	1 ⊢ ((𝐴 ∈ FinIa ∧ 𝑋 ⊆ 𝐴) → (𝑋 ∈ Fin ∨ (𝐴 ∖ 𝑋) ∈ Fin))

Syntax hints:   → wi 4   ∧ wa 394   ∨ wo 843   = wceq 1539   ∈ wcel 2104  ∀wral 3059   ∖ cdif 3946   ⊆ wss 3949  𝒫 cpw 4603  Fincfn 8943  Fin^Iacfin1a 10277

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5300

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rab 3431  df-v 3474  df-dif 3952  df-in 3956  df-ss 3966  df-pw 4605  df-fin1a 10284

This theorem is referenced by:  enfin1ai  10383  fin1a2  10414  fin1aufil  23658