Theorem ssfin2 10230

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 766	. . . 4 ⊢ (((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ 𝒫 𝒫 𝐵) → 𝐴 ∈ FinII)
2		elpwi 4561	. . . . . 6 ⊢ (𝑥 ∈ 𝒫 𝒫 𝐵 → 𝑥 ⊆ 𝒫 𝐵)
3	2	adantl 481	. . . . 5 ⊢ (((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ 𝒫 𝒫 𝐵) → 𝑥 ⊆ 𝒫 𝐵)
4		simplr 768	. . . . . 6 ⊢ (((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ 𝒫 𝒫 𝐵) → 𝐵 ⊆ 𝐴)
5	4	sspwd 4567	. . . . 5 ⊢ (((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ 𝒫 𝒫 𝐵) → 𝒫 𝐵 ⊆ 𝒫 𝐴)
6	3, 5	sstrd 3944	. . . 4 ⊢ (((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ 𝒫 𝒫 𝐵) → 𝑥 ⊆ 𝒫 𝐴)
7		fin2i 10205	. . . . 5 ⊢ (((𝐴 ∈ FinII ∧ 𝑥 ⊆ 𝒫 𝐴) ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥)) → ∪ 𝑥 ∈ 𝑥)
8	7	ex 412	. . . 4 ⊢ ((𝐴 ∈ FinII ∧ 𝑥 ⊆ 𝒫 𝐴) → ((𝑥 ≠ ∅ ∧ [⊊] Or 𝑥) → ∪ 𝑥 ∈ 𝑥))
9	1, 6, 8	syl2anc 584	. . 3 ⊢ (((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ 𝒫 𝒫 𝐵) → ((𝑥 ≠ ∅ ∧ [⊊] Or 𝑥) → ∪ 𝑥 ∈ 𝑥))
10	9	ralrimiva 3128	. 2 ⊢ ((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝐴) → ∀𝑥 ∈ 𝒫 𝒫 𝐵((𝑥 ≠ ∅ ∧ [⊊] Or 𝑥) → ∪ 𝑥 ∈ 𝑥))
11		ssexg 5268	. . . 4 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ∈ FinII) → 𝐵 ∈ V)
12	11	ancoms 458	. . 3 ⊢ ((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ V)
13		isfin2 10204	. . 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 257	1 ⊢ ((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ FinII)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2113   ≠ wne 2932  ∀wral 3051  Vcvv 3440   ⊆ wss 3901  ∅c0 4285  𝒫 cpw 4554  ∪ cuni 4863   Or wor 5531   [⊊] crpss 7667  Fin^IIcfin2 10189

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-pow 5310

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-in 3908  df-ss 3918  df-pw 4556  df-uni 4864  df-po 5532  df-so 5533  df-fin2 10196

This theorem is referenced by: (None)