Theorem ssnn0fi 13705

Description: A subset of the nonnegative integers is finite if and only if there is a nonnegative integer so that all integers greater than this integer are not contained in the subset. (Contributed by AV, 3-Oct-2019.)

Assertion

Ref	Expression
ssnn0fi	⊢ (𝑆 ⊆ ℕ0 → (𝑆 ∈ Fin ↔ ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)))

Distinct variable group:   𝑆,𝑠,𝑥

Proof of Theorem ssnn0fi

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0nn0 12248	. . . . . 6 ⊢ 0 ∈ ℕ0
2	1	a1i 11	. . . . 5 ⊢ (𝑆 = ∅ → 0 ∈ ℕ0)
3		breq1 5077	. . . . . . . 8 ⊢ (𝑠 = 0 → (𝑠 < 𝑥 ↔ 0 < 𝑥))
4	3	imbi1d 342	. . . . . . 7 ⊢ (𝑠 = 0 → ((𝑠 < 𝑥 → 𝑥 ∉ 𝑆) ↔ (0 < 𝑥 → 𝑥 ∉ 𝑆)))
5	4	ralbidv 3112	. . . . . 6 ⊢ (𝑠 = 0 → (∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆) ↔ ∀𝑥 ∈ ℕ0 (0 < 𝑥 → 𝑥 ∉ 𝑆)))
6	5	adantl 482	. . . . 5 ⊢ ((𝑆 = ∅ ∧ 𝑠 = 0) → (∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆) ↔ ∀𝑥 ∈ ℕ0 (0 < 𝑥 → 𝑥 ∉ 𝑆)))
7		nnel 3058	. . . . . . . . 9 ⊢ (¬ 𝑥 ∉ 𝑆 ↔ 𝑥 ∈ 𝑆)
8		n0i 4267	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑆 → ¬ 𝑆 = ∅)
9	7, 8	sylbi 216	. . . . . . . 8 ⊢ (¬ 𝑥 ∉ 𝑆 → ¬ 𝑆 = ∅)
10	9	con4i 114	. . . . . . 7 ⊢ (𝑆 = ∅ → 𝑥 ∉ 𝑆)
11	10	a1d 25	. . . . . 6 ⊢ (𝑆 = ∅ → (0 < 𝑥 → 𝑥 ∉ 𝑆))
12	11	ralrimivw 3104	. . . . 5 ⊢ (𝑆 = ∅ → ∀𝑥 ∈ ℕ0 (0 < 𝑥 → 𝑥 ∉ 𝑆))
13	2, 6, 12	rspcedvd 3563	. . . 4 ⊢ (𝑆 = ∅ → ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆))
14	13	2a1d 26	. . 3 ⊢ (𝑆 = ∅ → (𝑆 ⊆ ℕ0 → (𝑆 ∈ Fin → ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆))))
15		ltso 11055	. . . . . . 7 ⊢ < Or ℝ
16		id 22	. . . . . . . . 9 ⊢ (𝑆 ⊆ ℕ0 → 𝑆 ⊆ ℕ0)
17		nn0ssre 12237	. . . . . . . . 9 ⊢ ℕ0 ⊆ ℝ
18	16, 17	sstrdi 3933	. . . . . . . 8 ⊢ (𝑆 ⊆ ℕ0 → 𝑆 ⊆ ℝ)
19	18	3anim3i 1153	. . . . . . 7 ⊢ ((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) → (𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℝ))
20		fisup2g 9227	. . . . . . 7 ⊢ (( < Or ℝ ∧ (𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℝ)) → ∃𝑠 ∈ 𝑆 (∀𝑦 ∈ 𝑆 ¬ 𝑠 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑠 → ∃𝑧 ∈ 𝑆 𝑦 < 𝑧)))
21	15, 19, 20	sylancr 587	. . . . . 6 ⊢ ((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) → ∃𝑠 ∈ 𝑆 (∀𝑦 ∈ 𝑆 ¬ 𝑠 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑠 → ∃𝑧 ∈ 𝑆 𝑦 < 𝑧)))
22		simp3 1137	. . . . . . 7 ⊢ ((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) → 𝑆 ⊆ ℕ0)
23		breq2 5078	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑥 → (𝑠 < 𝑦 ↔ 𝑠 < 𝑥))
24	23	notbid 318	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑥 → (¬ 𝑠 < 𝑦 ↔ ¬ 𝑠 < 𝑥))
25	24	rspcva 3559	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 ¬ 𝑠 < 𝑦) → ¬ 𝑠 < 𝑥)
26	25	2a1d 26	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 ¬ 𝑠 < 𝑦) → (𝑥 ∈ ℕ0 → (((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) ∧ 𝑠 ∈ 𝑆) → ¬ 𝑠 < 𝑥)))
27	26	expcom 414	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑦 ∈ 𝑆 ¬ 𝑠 < 𝑦 → (𝑥 ∈ 𝑆 → (𝑥 ∈ ℕ0 → (((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) ∧ 𝑠 ∈ 𝑆) → ¬ 𝑠 < 𝑥))))
28	27	com24 95	. . . . . . . . . . . . . . 15 ⊢ (∀𝑦 ∈ 𝑆 ¬ 𝑠 < 𝑦 → (((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) ∧ 𝑠 ∈ 𝑆) → (𝑥 ∈ ℕ0 → (𝑥 ∈ 𝑆 → ¬ 𝑠 < 𝑥))))
29	28	imp31 418	. . . . . . . . . . . . . 14 ⊢ (((∀𝑦 ∈ 𝑆 ¬ 𝑠 < 𝑦 ∧ ((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) ∧ 𝑠 ∈ 𝑆)) ∧ 𝑥 ∈ ℕ0) → (𝑥 ∈ 𝑆 → ¬ 𝑠 < 𝑥))
30	7, 29	syl5bi 241	. . . . . . . . . . . . 13 ⊢ (((∀𝑦 ∈ 𝑆 ¬ 𝑠 < 𝑦 ∧ ((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) ∧ 𝑠 ∈ 𝑆)) ∧ 𝑥 ∈ ℕ0) → (¬ 𝑥 ∉ 𝑆 → ¬ 𝑠 < 𝑥))
31	30	con4d 115	. . . . . . . . . . . 12 ⊢ (((∀𝑦 ∈ 𝑆 ¬ 𝑠 < 𝑦 ∧ ((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) ∧ 𝑠 ∈ 𝑆)) ∧ 𝑥 ∈ ℕ0) → (𝑠 < 𝑥 → 𝑥 ∉ 𝑆))
32	31	ralrimiva 3103	. . . . . . . . . . 11 ⊢ ((∀𝑦 ∈ 𝑆 ¬ 𝑠 < 𝑦 ∧ ((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) ∧ 𝑠 ∈ 𝑆)) → ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆))
33	32	ex 413	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝑆 ¬ 𝑠 < 𝑦 → (((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) ∧ 𝑠 ∈ 𝑆) → ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)))
34	33	adantr 481	. . . . . . . . 9 ⊢ ((∀𝑦 ∈ 𝑆 ¬ 𝑠 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑠 → ∃𝑧 ∈ 𝑆 𝑦 < 𝑧)) → (((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) ∧ 𝑠 ∈ 𝑆) → ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)))
35	34	com12 32	. . . . . . . 8 ⊢ (((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) ∧ 𝑠 ∈ 𝑆) → ((∀𝑦 ∈ 𝑆 ¬ 𝑠 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑠 → ∃𝑧 ∈ 𝑆 𝑦 < 𝑧)) → ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)))
36	35	reximdva 3203	. . . . . . 7 ⊢ ((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) → (∃𝑠 ∈ 𝑆 (∀𝑦 ∈ 𝑆 ¬ 𝑠 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑠 → ∃𝑧 ∈ 𝑆 𝑦 < 𝑧)) → ∃𝑠 ∈ 𝑆 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)))
37		ssrexv 3988	. . . . . . 7 ⊢ (𝑆 ⊆ ℕ0 → (∃𝑠 ∈ 𝑆 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆) → ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)))
38	22, 36, 37	sylsyld 61	. . . . . 6 ⊢ ((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) → (∃𝑠 ∈ 𝑆 (∀𝑦 ∈ 𝑆 ¬ 𝑠 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑠 → ∃𝑧 ∈ 𝑆 𝑦 < 𝑧)) → ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)))
39	21, 38	mpd 15	. . . . 5 ⊢ ((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) → ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆))
40	39	3exp 1118	. . . 4 ⊢ (𝑆 ∈ Fin → (𝑆 ≠ ∅ → (𝑆 ⊆ ℕ0 → ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆))))
41	40	com3l 89	. . 3 ⊢ (𝑆 ≠ ∅ → (𝑆 ⊆ ℕ0 → (𝑆 ∈ Fin → ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆))))
42	14, 41	pm2.61ine 3028	. 2 ⊢ (𝑆 ⊆ ℕ0 → (𝑆 ∈ Fin → ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)))
43		fzfi 13692	. . . 4 ⊢ (0...𝑠) ∈ Fin
44		elfz2nn0 13347	. . . . . . . . . 10 ⊢ (𝑦 ∈ (0...𝑠) ↔ (𝑦 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0 ∧ 𝑦 ≤ 𝑠))
45	44	notbii 320	. . . . . . . . 9 ⊢ (¬ 𝑦 ∈ (0...𝑠) ↔ ¬ (𝑦 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0 ∧ 𝑦 ≤ 𝑠))
46		3ianor 1106	. . . . . . . . 9 ⊢ (¬ (𝑦 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0 ∧ 𝑦 ≤ 𝑠) ↔ (¬ 𝑦 ∈ ℕ0 ∨ ¬ 𝑠 ∈ ℕ0 ∨ ¬ 𝑦 ≤ 𝑠))
47		3orass 1089	. . . . . . . . 9 ⊢ ((¬ 𝑦 ∈ ℕ0 ∨ ¬ 𝑠 ∈ ℕ0 ∨ ¬ 𝑦 ≤ 𝑠) ↔ (¬ 𝑦 ∈ ℕ0 ∨ (¬ 𝑠 ∈ ℕ0 ∨ ¬ 𝑦 ≤ 𝑠)))
48	45, 46, 47	3bitri 297	. . . . . . . 8 ⊢ (¬ 𝑦 ∈ (0...𝑠) ↔ (¬ 𝑦 ∈ ℕ0 ∨ (¬ 𝑠 ∈ ℕ0 ∨ ¬ 𝑦 ≤ 𝑠)))
49		ssel 3914	. . . . . . . . . . . 12 ⊢ (𝑆 ⊆ ℕ0 → (𝑦 ∈ 𝑆 → 𝑦 ∈ ℕ0))
50	49	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) → (𝑦 ∈ 𝑆 → 𝑦 ∈ ℕ0))
51	50	adantr 481	. . . . . . . . . 10 ⊢ (((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → (𝑦 ∈ 𝑆 → 𝑦 ∈ ℕ0))
52	51	con3rr3 155	. . . . . . . . 9 ⊢ (¬ 𝑦 ∈ ℕ0 → (((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → ¬ 𝑦 ∈ 𝑆))
53		notnotb 315	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ0 ↔ ¬ ¬ 𝑦 ∈ ℕ0)
54		pm2.24 124	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ ℕ0 → (¬ 𝑠 ∈ ℕ0 → ¬ 𝑦 ∈ 𝑆))
55	54	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) → (¬ 𝑠 ∈ ℕ0 → ¬ 𝑦 ∈ 𝑆))
56	55	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → (¬ 𝑠 ∈ ℕ0 → ¬ 𝑦 ∈ 𝑆))
57	56	com12 32	. . . . . . . . . . . . 13 ⊢ (¬ 𝑠 ∈ ℕ0 → (((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → ¬ 𝑦 ∈ 𝑆))
58	57	a1d 25	. . . . . . . . . . . 12 ⊢ (¬ 𝑠 ∈ ℕ0 → (𝑦 ∈ ℕ0 → (((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → ¬ 𝑦 ∈ 𝑆)))
59		breq2 5078	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑦 → (𝑠 < 𝑥 ↔ 𝑠 < 𝑦))
60		neleq1 3054	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑦 → (𝑥 ∉ 𝑆 ↔ 𝑦 ∉ 𝑆))
61	59, 60	imbi12d 345	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → ((𝑠 < 𝑥 → 𝑥 ∉ 𝑆) ↔ (𝑠 < 𝑦 → 𝑦 ∉ 𝑆)))
62	61	rspcva 3559	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → (𝑠 < 𝑦 → 𝑦 ∉ 𝑆))
63		nn0re 12242	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑠 ∈ ℕ0 → 𝑠 ∈ ℝ)
64		nn0re 12242	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ ℕ0 → 𝑦 ∈ ℝ)
65		ltnle 11054	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑠 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑠 < 𝑦 ↔ ¬ 𝑦 ≤ 𝑠))
66	63, 64, 65	syl2an 596	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑠 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑠 < 𝑦 ↔ ¬ 𝑦 ≤ 𝑠))
67		df-nel 3050	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∉ 𝑆 ↔ ¬ 𝑦 ∈ 𝑆)
68	67	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑠 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑦 ∉ 𝑆 ↔ ¬ 𝑦 ∈ 𝑆))
69	66, 68	imbi12d 345	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑠 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → ((𝑠 < 𝑦 → 𝑦 ∉ 𝑆) ↔ (¬ 𝑦 ≤ 𝑠 → ¬ 𝑦 ∈ 𝑆)))
70	69	biimpd 228	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑠 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → ((𝑠 < 𝑦 → 𝑦 ∉ 𝑆) → (¬ 𝑦 ≤ 𝑠 → ¬ 𝑦 ∈ 𝑆)))
71	70	ex 413	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑠 ∈ ℕ0 → (𝑦 ∈ ℕ0 → ((𝑠 < 𝑦 → 𝑦 ∉ 𝑆) → (¬ 𝑦 ≤ 𝑠 → ¬ 𝑦 ∈ 𝑆))))
72	71	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) → (𝑦 ∈ ℕ0 → ((𝑠 < 𝑦 → 𝑦 ∉ 𝑆) → (¬ 𝑦 ≤ 𝑠 → ¬ 𝑦 ∈ 𝑆))))
73	72	com12 32	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ℕ0 → ((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) → ((𝑠 < 𝑦 → 𝑦 ∉ 𝑆) → (¬ 𝑦 ≤ 𝑠 → ¬ 𝑦 ∈ 𝑆))))
74	73	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → ((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) → ((𝑠 < 𝑦 → 𝑦 ∉ 𝑆) → (¬ 𝑦 ≤ 𝑠 → ¬ 𝑦 ∈ 𝑆))))
75	62, 74	mpid 44	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → ((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) → (¬ 𝑦 ≤ 𝑠 → ¬ 𝑦 ∈ 𝑆)))
76	75	ex 413	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℕ0 → (∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆) → ((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) → (¬ 𝑦 ≤ 𝑠 → ¬ 𝑦 ∈ 𝑆))))
77	76	com13 88	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) → (∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆) → (𝑦 ∈ ℕ0 → (¬ 𝑦 ≤ 𝑠 → ¬ 𝑦 ∈ 𝑆))))
78	77	imp 407	. . . . . . . . . . . . 13 ⊢ (((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → (𝑦 ∈ ℕ0 → (¬ 𝑦 ≤ 𝑠 → ¬ 𝑦 ∈ 𝑆)))
79	78	com13 88	. . . . . . . . . . . 12 ⊢ (¬ 𝑦 ≤ 𝑠 → (𝑦 ∈ ℕ0 → (((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → ¬ 𝑦 ∈ 𝑆)))
80	58, 79	jaoi 854	. . . . . . . . . . 11 ⊢ ((¬ 𝑠 ∈ ℕ0 ∨ ¬ 𝑦 ≤ 𝑠) → (𝑦 ∈ ℕ0 → (((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → ¬ 𝑦 ∈ 𝑆)))
81	53, 80	syl5bir 242	. . . . . . . . . 10 ⊢ ((¬ 𝑠 ∈ ℕ0 ∨ ¬ 𝑦 ≤ 𝑠) → (¬ ¬ 𝑦 ∈ ℕ0 → (((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → ¬ 𝑦 ∈ 𝑆)))
82	81	impcom 408	. . . . . . . . 9 ⊢ ((¬ ¬ 𝑦 ∈ ℕ0 ∧ (¬ 𝑠 ∈ ℕ0 ∨ ¬ 𝑦 ≤ 𝑠)) → (((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → ¬ 𝑦 ∈ 𝑆))
83	52, 82	jaoi3 1058	. . . . . . . 8 ⊢ ((¬ 𝑦 ∈ ℕ0 ∨ (¬ 𝑠 ∈ ℕ0 ∨ ¬ 𝑦 ≤ 𝑠)) → (((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → ¬ 𝑦 ∈ 𝑆))
84	48, 83	sylbi 216	. . . . . . 7 ⊢ (¬ 𝑦 ∈ (0...𝑠) → (((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → ¬ 𝑦 ∈ 𝑆))
85	84	com12 32	. . . . . 6 ⊢ (((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → (¬ 𝑦 ∈ (0...𝑠) → ¬ 𝑦 ∈ 𝑆))
86	85	con4d 115	. . . . 5 ⊢ (((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → (𝑦 ∈ 𝑆 → 𝑦 ∈ (0...𝑠)))
87	86	ssrdv 3927	. . . 4 ⊢ (((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → 𝑆 ⊆ (0...𝑠))
88		ssfi 8956	. . . 4 ⊢ (((0...𝑠) ∈ Fin ∧ 𝑆 ⊆ (0...𝑠)) → 𝑆 ∈ Fin)
89	43, 87, 88	sylancr 587	. . 3 ⊢ (((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → 𝑆 ∈ Fin)
90	89	rexlimdva2 3216	. 2 ⊢ (𝑆 ⊆ ℕ0 → (∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆) → 𝑆 ∈ Fin))
91	42, 90	impbid 211	1 ⊢ (𝑆 ⊆ ℕ0 → (𝑆 ∈ Fin ↔ ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844   ∨ w3o 1085   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   ≠ wne 2943   ∉ wnel 3049  ∀wral 3064  ∃wrex 3065   ⊆ wss 3887  ∅c0 4256   class class class wbr 5074   Or wor 5502  (class class class)co 7275  Fincfn 8733  ℝcr 10870  0cc0 10871   < clt 11009   ≤ cle 11010  ℕ₀cn0 12233  ...cfz 13239

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-n0 12234  df-z 12320  df-uz 12583  df-fz 13240

This theorem is referenced by:  rabssnn0fi  13706