Theorem fin2i 10224

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

Assertion

Ref	Expression
fin2i	⊢ (((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝒫 𝐴) ∧ (𝐵 ≠ ∅ ∧ [⊊] Or 𝐵)) → ∪ 𝐵 ∈ 𝐵)

Proof of Theorem fin2i

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		neeq1 2987	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 ≠ ∅ ↔ 𝐵 ≠ ∅))
2		soeq2 5561	. . . . 5 ⊢ (𝑦 = 𝐵 → ( [⊊] Or 𝑦 ↔ [⊊] Or 𝐵))
3	1, 2	anbi12d 632	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) ↔ (𝐵 ≠ ∅ ∧ [⊊] Or 𝐵)))
4		unieq 4878	. . . . 5 ⊢ (𝑦 = 𝐵 → ∪ 𝑦 = ∪ 𝐵)
5		id 22	. . . . 5 ⊢ (𝑦 = 𝐵 → 𝑦 = 𝐵)
6	4, 5	eleq12d 2822	. . . 4 ⊢ (𝑦 = 𝐵 → (∪ 𝑦 ∈ 𝑦 ↔ ∪ 𝐵 ∈ 𝐵))
7	3, 6	imbi12d 344	. . 3 ⊢ (𝑦 = 𝐵 → (((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∪ 𝑦 ∈ 𝑦) ↔ ((𝐵 ≠ ∅ ∧ [⊊] Or 𝐵) → ∪ 𝐵 ∈ 𝐵)))
8		isfin2 10223	. . . . 5 ⊢ (𝐴 ∈ FinII → (𝐴 ∈ FinII ↔ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∪ 𝑦 ∈ 𝑦)))
9	8	ibi 267	. . . 4 ⊢ (𝐴 ∈ FinII → ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∪ 𝑦 ∈ 𝑦))
10	9	adantr 480	. . 3 ⊢ ((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝒫 𝐴) → ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∪ 𝑦 ∈ 𝑦))
11		pwexg 5328	. . . . 5 ⊢ (𝐴 ∈ FinII → 𝒫 𝐴 ∈ V)
12		elpw2g 5283	. . . . 5 ⊢ (𝒫 𝐴 ∈ V → (𝐵 ∈ 𝒫 𝒫 𝐴 ↔ 𝐵 ⊆ 𝒫 𝐴))
13	11, 12	syl 17	. . . 4 ⊢ (𝐴 ∈ FinII → (𝐵 ∈ 𝒫 𝒫 𝐴 ↔ 𝐵 ⊆ 𝒫 𝐴))
14	13	biimpar 477	. . 3 ⊢ ((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝒫 𝐴) → 𝐵 ∈ 𝒫 𝒫 𝐴)
15	7, 10, 14	rspcdva 3586	. 2 ⊢ ((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝒫 𝐴) → ((𝐵 ≠ ∅ ∧ [⊊] Or 𝐵) → ∪ 𝐵 ∈ 𝐵))
16	15	imp 406	1 ⊢ (((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝒫 𝐴) ∧ (𝐵 ≠ ∅ ∧ [⊊] Or 𝐵)) → ∪ 𝐵 ∈ 𝐵)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  Vcvv 3444   ⊆ wss 3911  ∅c0 4292  𝒫 cpw 4559  ∪ cuni 4867   Or wor 5538   [⊊] crpss 7678  Fin^IIcfin2 10208

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5246  ax-pow 5315

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-in 3918  df-ss 3928  df-pw 4561  df-uni 4868  df-po 5539  df-so 5540  df-fin2 10215

This theorem is referenced by:  fin2i2  10247  ssfin2  10249  enfin2i  10250  fin1a2lem13  10341