Theorem fin1ai 10252

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 2852	. . 3 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ Fin ↔ 𝑋 ∈ Fin))
2		difeq2 4076	. . . 4 ⊢ (𝑥 = 𝑋 → (𝐴 ∖ 𝑥) = (𝐴 ∖ 𝑋))
3	2	eleq1d 2849	. . 3 ⊢ (𝑥 = 𝑋 → ((𝐴 ∖ 𝑥) ∈ Fin ↔ (𝐴 ∖ 𝑋) ∈ Fin))
4	1, 3	orbi12d 929	. 2 ⊢ (𝑥 = 𝑋 → ((𝑥 ∈ Fin ∨ (𝐴 ∖ 𝑥) ∈ Fin) ↔ (𝑋 ∈ Fin ∨ (𝐴 ∖ 𝑋) ∈ Fin)))
5		isfin1a 10251	. . . 4 ⊢ (𝐴 ∈ FinIa → (𝐴 ∈ FinIa ↔ ∀𝑥 ∈ 𝒫 𝐴(𝑥 ∈ Fin ∨ (𝐴 ∖ 𝑥) ∈ Fin)))
6	5	ibi 269	. . 3 ⊢ (𝐴 ∈ FinIa → ∀𝑥 ∈ 𝒫 𝐴(𝑥 ∈ Fin ∨ (𝐴 ∖ 𝑥) ∈ Fin))
7	6	adantr 484	. 2 ⊢ ((𝐴 ∈ FinIa ∧ 𝑋 ⊆ 𝐴) → ∀𝑥 ∈ 𝒫 𝐴(𝑥 ∈ Fin ∨ (𝐴 ∖ 𝑥) ∈ Fin))
8		elpw2g 5291	. . 3 ⊢ (𝐴 ∈ FinIa → (𝑋 ∈ 𝒫 𝐴 ↔ 𝑋 ⊆ 𝐴))
9	8	biimpar 481	. 2 ⊢ ((𝐴 ∈ FinIa ∧ 𝑋 ⊆ 𝐴) → 𝑋 ∈ 𝒫 𝐴)
10	4, 7, 9	rspcdva 3584	1 ⊢ ((𝐴 ∈ FinIa ∧ 𝑋 ⊆ 𝐴) → (𝑋 ∈ Fin ∨ (𝐴 ∖ 𝑋) ∈ Fin))

Syntax hints:   → wi 4   ∧ wa 399   ∨ wo 858   = wceq 1562   ∈ wcel 2144  ∀wral 3078   ∖ cdif 3903   ⊆ wss 3906  𝒫 cpw 4557  Fincfn 8929  Fin^Iacfin1a 10237

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736  ax-sep 5248

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ral 3079  df-rab 3417  df-v 3458  df-dif 3909  df-in 3913  df-ss 3923  df-pw 4559  df-fin1a 10244

This theorem is referenced by:  enfin1ai  10343  fin1a2  10374  fin1aufil  23994