Theorem fin1ai 10371

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 2848	. . 3 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ Fin ↔ 𝑋 ∈ Fin))
2		difeq2 4057	. . . 4 ⊢ (𝑥 = 𝑋 → (𝐴 ∖ 𝑥) = (𝐴 ∖ 𝑋))
3	2	eleq1d 2845	. . 3 ⊢ (𝑥 = 𝑋 → ((𝐴 ∖ 𝑥) ∈ Fin ↔ (𝐴 ∖ 𝑋) ∈ Fin))
4	1, 3	orbi12d 932	. 2 ⊢ (𝑥 = 𝑋 → ((𝑥 ∈ Fin ∨ (𝐴 ∖ 𝑥) ∈ Fin) ↔ (𝑋 ∈ Fin ∨ (𝐴 ∖ 𝑋) ∈ Fin)))
5		isfin1a 10370	. . . 4 ⊢ (𝐴 ∈ FinIa → (𝐴 ∈ FinIa ↔ ∀𝑥 ∈ 𝒫 𝐴(𝑥 ∈ Fin ∨ (𝐴 ∖ 𝑥) ∈ Fin)))
6	5	ibi 270	. . 3 ⊢ (𝐴 ∈ FinIa → ∀𝑥 ∈ 𝒫 𝐴(𝑥 ∈ Fin ∨ (𝐴 ∖ 𝑥) ∈ Fin))
7	6	adantr 486	. 2 ⊢ ((𝐴 ∈ FinIa ∧ 𝑋 ⊆ 𝐴) → ∀𝑥 ∈ 𝒫 𝐴(𝑥 ∈ Fin ∨ (𝐴 ∖ 𝑥) ∈ Fin))
8		elpw2g 5294	. . 3 ⊢ (𝐴 ∈ FinIa → (𝑋 ∈ 𝒫 𝐴 ↔ 𝑋 ⊆ 𝐴))
9	8	biimpar 483	. 2 ⊢ ((𝐴 ∈ FinIa ∧ 𝑋 ⊆ 𝐴) → 𝑋 ∈ 𝒫 𝐴)
10	4, 7, 9	rspcdva 3577	1 ⊢ ((𝐴 ∈ FinIa ∧ 𝑋 ⊆ 𝐴) → (𝑋 ∈ Fin ∨ (𝐴 ∖ 𝑋) ∈ Fin))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   = wceq 1569   ∈ wcel 2144  ∀wral 3076   ⊆ wss 3895   ∖ cdif 4045  𝒫 cpw 4556  Fincfn 8973  Fin^Iacfin1a 10356

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-pw 4558  df-fin1a 10363

This theorem is used by:  enfin1ai  10462  fin1a2  10493  fin1aufil  24251