Theorem ssnn0fi 12784

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 11307	. . . . . 6 ⊢ 0 ∈ ℕ0
2	1	a1i 11	. . . . 5 ⊢ (𝑆 = ∅ → 0 ∈ ℕ0)
3		breq1 4656	. . . . . . . 8 ⊢ (𝑠 = 0 → (𝑠 < 𝑥 ↔ 0 < 𝑥))
4	3	imbi1d 331	. . . . . . 7 ⊢ (𝑠 = 0 → ((𝑠 < 𝑥 → 𝑥 ∉ 𝑆) ↔ (0 < 𝑥 → 𝑥 ∉ 𝑆)))
5	4	ralbidv 2986	. . . . . 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 2906	. . . . . . . . 9 ⊢ (¬ 𝑥 ∉ 𝑆 ↔ 𝑥 ∈ 𝑆)
8		n0i 3920	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑆 → ¬ 𝑆 = ∅)
9	7, 8	sylbi 207	. . . . . . . 8 ⊢ (¬ 𝑥 ∉ 𝑆 → ¬ 𝑆 = ∅)
10	9	con4i 113	. . . . . . 7 ⊢ (𝑆 = ∅ → 𝑥 ∉ 𝑆)
11	10	a1d 25	. . . . . 6 ⊢ (𝑆 = ∅ → (0 < 𝑥 → 𝑥 ∉ 𝑆))
12	11	ralrimivw 2967	. . . . 5 ⊢ (𝑆 = ∅ → ∀𝑥 ∈ ℕ0 (0 < 𝑥 → 𝑥 ∉ 𝑆))
13	2, 6, 12	rspcedvd 3317	. . . 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 10118	. . . . . . 7 ⊢ < Or ℝ
16		id 22	. . . . . . . . 9 ⊢ (𝑆 ⊆ ℕ0 → 𝑆 ⊆ ℕ0)
17		nn0ssre 11296	. . . . . . . . 9 ⊢ ℕ0 ⊆ ℝ
18	16, 17	syl6ss 3615	. . . . . . . 8 ⊢ (𝑆 ⊆ ℕ0 → 𝑆 ⊆ ℝ)
19	18	3anim3i 1250	. . . . . . 7 ⊢ ((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) → (𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℝ))
20		fisup2g 8374	. . . . . . 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 695	. . . . . 6 ⊢ ((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) → ∃𝑠 ∈ 𝑆 (∀𝑦 ∈ 𝑆 ¬ 𝑠 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑠 → ∃𝑧 ∈ 𝑆 𝑦 < 𝑧)))
22		simp3 1063	. . . . . . 7 ⊢ ((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) → 𝑆 ⊆ ℕ0)
23		breq2 4657	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑥 → (𝑠 < 𝑦 ↔ 𝑠 < 𝑥))
24	23	notbid 308	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑥 → (¬ 𝑠 < 𝑦 ↔ ¬ 𝑠 < 𝑥))
25	24	rspcva 3307	. . . . . . . . . . . . . . . . . 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 451	. . . . . . . . . . . . . . . 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 448	. . . . . . . . . . . . . 14 ⊢ (((∀𝑦 ∈ 𝑆 ¬ 𝑠 < 𝑦 ∧ ((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) ∧ 𝑠 ∈ 𝑆)) ∧ 𝑥 ∈ ℕ0) → (𝑥 ∈ 𝑆 → ¬ 𝑠 < 𝑥))
30	7, 29	syl5bi 232	. . . . . . . . . . . . 13 ⊢ (((∀𝑦 ∈ 𝑆 ¬ 𝑠 < 𝑦 ∧ ((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) ∧ 𝑠 ∈ 𝑆)) ∧ 𝑥 ∈ ℕ0) → (¬ 𝑥 ∉ 𝑆 → ¬ 𝑠 < 𝑥))
31	30	con4d 114	. . . . . . . . . . . 12 ⊢ (((∀𝑦 ∈ 𝑆 ¬ 𝑠 < 𝑦 ∧ ((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) ∧ 𝑠 ∈ 𝑆)) ∧ 𝑥 ∈ ℕ0) → (𝑠 < 𝑥 → 𝑥 ∉ 𝑆))
32	31	ralrimiva 2966	. . . . . . . . . . 11 ⊢ ((∀𝑦 ∈ 𝑆 ¬ 𝑠 < 𝑦 ∧ ((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) ∧ 𝑠 ∈ 𝑆)) → ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆))
33	32	ex 450	. . . . . . . . . 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 3017	. . . . . . 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 3667	. . . . . . 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 1264	. . . 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 2877	. 2 ⊢ (𝑆 ⊆ ℕ0 → (𝑆 ∈ Fin → ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)))
43		fzfi 12771	. . . . 5 ⊢ (0...𝑠) ∈ Fin
44		elfz2nn0 12431	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (0...𝑠) ↔ (𝑦 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0 ∧ 𝑦 ≤ 𝑠))
45	44	notbii 310	. . . . . . . . . 10 ⊢ (¬ 𝑦 ∈ (0...𝑠) ↔ ¬ (𝑦 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0 ∧ 𝑦 ≤ 𝑠))
46		3ianor 1055	. . . . . . . . . 10 ⊢ (¬ (𝑦 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0 ∧ 𝑦 ≤ 𝑠) ↔ (¬ 𝑦 ∈ ℕ0 ∨ ¬ 𝑠 ∈ ℕ0 ∨ ¬ 𝑦 ≤ 𝑠))
47		3orass 1040	. . . . . . . . . 10 ⊢ ((¬ 𝑦 ∈ ℕ0 ∨ ¬ 𝑠 ∈ ℕ0 ∨ ¬ 𝑦 ≤ 𝑠) ↔ (¬ 𝑦 ∈ ℕ0 ∨ (¬ 𝑠 ∈ ℕ0 ∨ ¬ 𝑦 ≤ 𝑠)))
48	45, 46, 47	3bitri 286	. . . . . . . . 9 ⊢ (¬ 𝑦 ∈ (0...𝑠) ↔ (¬ 𝑦 ∈ ℕ0 ∨ (¬ 𝑠 ∈ ℕ0 ∨ ¬ 𝑦 ≤ 𝑠)))
49		ssel 3597	. . . . . . . . . . . . 13 ⊢ (𝑆 ⊆ ℕ0 → (𝑦 ∈ 𝑆 → 𝑦 ∈ ℕ0))
50	49	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) → (𝑦 ∈ 𝑆 → 𝑦 ∈ ℕ0))
51	50	adantr 481	. . . . . . . . . . 11 ⊢ (((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → (𝑦 ∈ 𝑆 → 𝑦 ∈ ℕ0))
52	51	con3rr3 151	. . . . . . . . . 10 ⊢ (¬ 𝑦 ∈ ℕ0 → (((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → ¬ 𝑦 ∈ 𝑆))
53		notnotb 304	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℕ0 ↔ ¬ ¬ 𝑦 ∈ ℕ0)
54		pm2.24 121	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 ∈ ℕ0 → (¬ 𝑠 ∈ ℕ0 → ¬ 𝑦 ∈ 𝑆))
55	54	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) → (¬ 𝑠 ∈ ℕ0 → ¬ 𝑦 ∈ 𝑆))
56	55	adantr 481	. . . . . . . . . . . . . . 15 ⊢ (((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → (¬ 𝑠 ∈ ℕ0 → ¬ 𝑦 ∈ 𝑆))
57	56	com12 32	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑠 ∈ ℕ0 → (((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → ¬ 𝑦 ∈ 𝑆))
58	57	a1d 25	. . . . . . . . . . . . 13 ⊢ (¬ 𝑠 ∈ ℕ0 → (𝑦 ∈ ℕ0 → (((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → ¬ 𝑦 ∈ 𝑆)))
59		breq2 4657	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑦 → (𝑠 < 𝑥 ↔ 𝑠 < 𝑦))
60		neleq1 2902	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑦 → (𝑥 ∉ 𝑆 ↔ 𝑦 ∉ 𝑆))
61	59, 60	imbi12d 334	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑦 → ((𝑠 < 𝑥 → 𝑥 ∉ 𝑆) ↔ (𝑠 < 𝑦 → 𝑦 ∉ 𝑆)))
62	61	rspcva 3307	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → (𝑠 < 𝑦 → 𝑦 ∉ 𝑆))
63		nn0re 11301	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑠 ∈ ℕ0 → 𝑠 ∈ ℝ)
64		nn0re 11301	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∈ ℕ0 → 𝑦 ∈ ℝ)
65		ltnle 10117	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑠 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑠 < 𝑦 ↔ ¬ 𝑦 ≤ 𝑠))
66	63, 64, 65	syl2an 494	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑠 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑠 < 𝑦 ↔ ¬ 𝑦 ≤ 𝑠))
67		df-nel 2898	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∉ 𝑆 ↔ ¬ 𝑦 ∈ 𝑆)
68	67	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑠 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑦 ∉ 𝑆 ↔ ¬ 𝑦 ∈ 𝑆))
69	66, 68	imbi12d 334	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑠 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → ((𝑠 < 𝑦 → 𝑦 ∉ 𝑆) ↔ (¬ 𝑦 ≤ 𝑠 → ¬ 𝑦 ∈ 𝑆)))
70	69	biimpd 219	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑠 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → ((𝑠 < 𝑦 → 𝑦 ∉ 𝑆) → (¬ 𝑦 ≤ 𝑠 → ¬ 𝑦 ∈ 𝑆)))
71	70	ex 450	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑠 ∈ ℕ0 → (𝑦 ∈ ℕ0 → ((𝑠 < 𝑦 → 𝑦 ∉ 𝑆) → (¬ 𝑦 ≤ 𝑠 → ¬ 𝑦 ∈ 𝑆))))
72	71	adantl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) → (𝑦 ∈ ℕ0 → ((𝑠 < 𝑦 → 𝑦 ∉ 𝑆) → (¬ 𝑦 ≤ 𝑠 → ¬ 𝑦 ∈ 𝑆))))
73	72	com12 32	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ℕ0 → ((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) → ((𝑠 < 𝑦 → 𝑦 ∉ 𝑆) → (¬ 𝑦 ≤ 𝑠 → ¬ 𝑦 ∈ 𝑆))))
74	73	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((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	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → ((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) → (¬ 𝑦 ≤ 𝑠 → ¬ 𝑦 ∈ 𝑆)))
76	75	ex 450	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℕ0 → (∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆) → ((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) → (¬ 𝑦 ≤ 𝑠 → ¬ 𝑦 ∈ 𝑆))))
77	76	com13 88	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) → (∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆) → (𝑦 ∈ ℕ0 → (¬ 𝑦 ≤ 𝑠 → ¬ 𝑦 ∈ 𝑆))))
78	77	imp 445	. . . . . . . . . . . . . 14 ⊢ (((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → (𝑦 ∈ ℕ0 → (¬ 𝑦 ≤ 𝑠 → ¬ 𝑦 ∈ 𝑆)))
79	78	com13 88	. . . . . . . . . . . . 13 ⊢ (¬ 𝑦 ≤ 𝑠 → (𝑦 ∈ ℕ0 → (((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → ¬ 𝑦 ∈ 𝑆)))
80	58, 79	jaoi 394	. . . . . . . . . . . 12 ⊢ ((¬ 𝑠 ∈ ℕ0 ∨ ¬ 𝑦 ≤ 𝑠) → (𝑦 ∈ ℕ0 → (((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → ¬ 𝑦 ∈ 𝑆)))
81	53, 80	syl5bir 233	. . . . . . . . . . 11 ⊢ ((¬ 𝑠 ∈ ℕ0 ∨ ¬ 𝑦 ≤ 𝑠) → (¬ ¬ 𝑦 ∈ ℕ0 → (((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → ¬ 𝑦 ∈ 𝑆)))
82	81	impcom 446	. . . . . . . . . 10 ⊢ ((¬ ¬ 𝑦 ∈ ℕ0 ∧ (¬ 𝑠 ∈ ℕ0 ∨ ¬ 𝑦 ≤ 𝑠)) → (((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → ¬ 𝑦 ∈ 𝑆))
83	52, 82	jaoi3 1011	. . . . . . . . 9 ⊢ ((¬ 𝑦 ∈ ℕ0 ∨ (¬ 𝑠 ∈ ℕ0 ∨ ¬ 𝑦 ≤ 𝑠)) → (((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → ¬ 𝑦 ∈ 𝑆))
84	48, 83	sylbi 207	. . . . . . . 8 ⊢ (¬ 𝑦 ∈ (0...𝑠) → (((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → ¬ 𝑦 ∈ 𝑆))
85	84	com12 32	. . . . . . 7 ⊢ (((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → (¬ 𝑦 ∈ (0...𝑠) → ¬ 𝑦 ∈ 𝑆))
86	85	con4d 114	. . . . . 6 ⊢ (((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → (𝑦 ∈ 𝑆 → 𝑦 ∈ (0...𝑠)))
87	86	ssrdv 3609	. . . . 5 ⊢ (((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → 𝑆 ⊆ (0...𝑠))
88		ssfi 8180	. . . . 5 ⊢ (((0...𝑠) ∈ Fin ∧ 𝑆 ⊆ (0...𝑠)) → 𝑆 ∈ Fin)
89	43, 87, 88	sylancr 695	. . . 4 ⊢ (((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → 𝑆 ∈ Fin)
90	89	ex 450	. . 3 ⊢ ((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) → (∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆) → 𝑆 ∈ Fin))
91	90	rexlimdva 3031	. 2 ⊢ (𝑆 ⊆ ℕ0 → (∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆) → 𝑆 ∈ Fin))
92	42, 91	impbid 202	1 ⊢ (𝑆 ⊆ ℕ0 → (𝑆 ∈ Fin ↔ ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   ∨ w3o 1036   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794   ∉ wnel 2897  ∀wral 2912  ∃wrex 2913   ⊆ wss 3574  ∅c0 3915   class class class wbr 4653   Or wor 5034  (class class class)co 6650  Fincfn 7955  ℝcr 9935  0cc0 9936   < clt 10074   ≤ cle 10075  ℕ₀cn0 11292  ...cfz 12326

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327

This theorem is referenced by:  rabssnn0fi  12785