Theorem ssfin2 9934

Description: A subset of a II-finite set is II-finite. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Revised by Mario Carneiro, 16-May-2015.)

Assertion

Ref	Expression
ssfin2	⊢ ((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ FinII)

Proof of Theorem ssfin2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpll 767	. . . 4 ⊢ (((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ 𝒫 𝒫 𝐵) → 𝐴 ∈ FinII)
2		elpwi 4522	. . . . . 6 ⊢ (𝑥 ∈ 𝒫 𝒫 𝐵 → 𝑥 ⊆ 𝒫 𝐵)
3	2	adantl 485	. . . . 5 ⊢ (((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ 𝒫 𝒫 𝐵) → 𝑥 ⊆ 𝒫 𝐵)
4		simplr 769	. . . . . 6 ⊢ (((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ 𝒫 𝒫 𝐵) → 𝐵 ⊆ 𝐴)
5	4	sspwd 4528	. . . . 5 ⊢ (((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ 𝒫 𝒫 𝐵) → 𝒫 𝐵 ⊆ 𝒫 𝐴)
6	3, 5	sstrd 3911	. . . 4 ⊢ (((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ 𝒫 𝒫 𝐵) → 𝑥 ⊆ 𝒫 𝐴)
7		fin2i 9909	. . . . 5 ⊢ (((𝐴 ∈ FinII ∧ 𝑥 ⊆ 𝒫 𝐴) ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥)) → ∪ 𝑥 ∈ 𝑥)
8	7	ex 416	. . . 4 ⊢ ((𝐴 ∈ FinII ∧ 𝑥 ⊆ 𝒫 𝐴) → ((𝑥 ≠ ∅ ∧ [⊊] Or 𝑥) → ∪ 𝑥 ∈ 𝑥))
9	1, 6, 8	syl2anc 587	. . 3 ⊢ (((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ 𝒫 𝒫 𝐵) → ((𝑥 ≠ ∅ ∧ [⊊] Or 𝑥) → ∪ 𝑥 ∈ 𝑥))
10	9	ralrimiva 3105	. 2 ⊢ ((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝐴) → ∀𝑥 ∈ 𝒫 𝒫 𝐵((𝑥 ≠ ∅ ∧ [⊊] Or 𝑥) → ∪ 𝑥 ∈ 𝑥))
11		ssexg 5216	. . . 4 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ∈ FinII) → 𝐵 ∈ V)
12	11	ancoms 462	. . 3 ⊢ ((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ V)
13		isfin2 9908	. . 3 ⊢ (𝐵 ∈ V → (𝐵 ∈ FinII ↔ ∀𝑥 ∈ 𝒫 𝒫 𝐵((𝑥 ≠ ∅ ∧ [⊊] Or 𝑥) → ∪ 𝑥 ∈ 𝑥)))
14	12, 13	syl 17	. 2 ⊢ ((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝐴) → (𝐵 ∈ FinII ↔ ∀𝑥 ∈ 𝒫 𝒫 𝐵((𝑥 ≠ ∅ ∧ [⊊] Or 𝑥) → ∪ 𝑥 ∈ 𝑥)))
15	10, 14	mpbird 260	1 ⊢ ((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ FinII)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∈ wcel 2110   ≠ wne 2940  ∀wral 3061  Vcvv 3408   ⊆ wss 3866  ∅c0 4237  𝒫 cpw 4513  ∪ cuni 4819   Or wor 5467   [⊊] crpss 7510  Fin^IIcfin2 9893

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708  ax-sep 5192  ax-pow 5258

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3066  df-rab 3070  df-v 3410  df-in 3873  df-ss 3883  df-pw 4515  df-uni 4820  df-po 5468  df-so 5469  df-fin2 9900

This theorem is referenced by: (None)