Theorem fin1ai 10342

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 4067	. . . 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 10341	. . . 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 1570   ∈ wcel 2145  ∀wral 3076   ∖ cdif 3895   ⊆ wss 3898  𝒫 cpw 4556  Fincfn 8951  Fin^Iacfin1a 10327

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  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 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rab 3413  df-v 3452  df-dif 3901  df-in 3905  df-ss 3915  df-pw 4558  df-fin1a 10334

This theorem is used by:  enfin1ai  10433  fin1a2  10464  fin1aufil  24212