Theorem fin2i 10366

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 3018	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 ≠ ∅ ↔ 𝐵 ≠ ∅))
2		soeq2 5581	. . . . 5 ⊢ (𝑦 = 𝐵 → ( [⊊] Or 𝑦 ↔ [⊊] Or 𝐵))
3	1, 2	anbi12d 644	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) ↔ (𝐵 ≠ ∅ ∧ [⊊] Or 𝐵)))
4		unieq 4878	. . . . 5 ⊢ (𝑦 = 𝐵 → ∪ 𝑦 = ∪ 𝐵)
5		id 23	. . . . 5 ⊢ (𝑦 = 𝐵 → 𝑦 = 𝐵)
6	4, 5	eleq12d 2855	. . . 4 ⊢ (𝑦 = 𝐵 → (∪ 𝑦 ∈ 𝑦 ↔ ∪ 𝐵 ∈ 𝐵))
7	3, 6	imbi12d 347	. . 3 ⊢ (𝑦 = 𝐵 → (((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∪ 𝑦 ∈ 𝑦) ↔ ((𝐵 ≠ ∅ ∧ [⊊] Or 𝐵) → ∪ 𝐵 ∈ 𝐵)))
8		isfin2 10365	. . . . 5 ⊢ (𝐴 ∈ FinII → (𝐴 ∈ FinII ↔ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∪ 𝑦 ∈ 𝑦)))
9	8	ibi 270	. . . 4 ⊢ (𝐴 ∈ FinII → ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∪ 𝑦 ∈ 𝑦))
10	9	adantr 486	. . 3 ⊢ ((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝒫 𝐴) → ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∪ 𝑦 ∈ 𝑦))
11		pwexg 5340	. . . . 5 ⊢ (𝐴 ∈ FinII → 𝒫 𝐴 ∈ V)
12		elpw2g 5295	. . . . 5 ⊢ (𝒫 𝐴 ∈ V → (𝐵 ∈ 𝒫 𝒫 𝐴 ↔ 𝐵 ⊆ 𝒫 𝐴))
13	11, 12	syl 18	. . . 4 ⊢ (𝐴 ∈ FinII → (𝐵 ∈ 𝒫 𝒫 𝐴 ↔ 𝐵 ⊆ 𝒫 𝐴))
14	13	biimpar 483	. . 3 ⊢ ((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝒫 𝐴) → 𝐵 ∈ 𝒫 𝒫 𝐴)
15	7, 10, 14	rspcdva 3578	. 2 ⊢ ((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝒫 𝐴) → ((𝐵 ≠ ∅ ∧ [⊊] Or 𝐵) → ∪ 𝐵 ∈ 𝐵))
16	15	imp 412	1 ⊢ (((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝒫 𝐴) ∧ (𝐵 ≠ ∅ ∧ [⊊] Or 𝐵)) → ∪ 𝐵 ∈ 𝐵)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∀wral 3077  Vcvv 3451   ⊆ wss 3899  ∅c0 4279  𝒫 cpw 4557  ∪ cuni 4867   Or wor 5558   [⊊] crpss 7736  Fin^IIcfin2 10350

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 2733  ax-sep 5249  ax-pow 5327

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-in 3906  df-ss 3916  df-pw 4559  df-uni 4868  df-po 5559  df-so 5560  df-fin2 10357

This theorem is used by:  fin2i2  10389  ssfin2  10391  enfin2i  10392  fin1a2lem13  10483