Theorem ssnn0fi 13356

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 11915	. . . . . 6 ⊢ 0 ∈ ℕ0
2	1	a1i 11	. . . . 5 ⊢ (𝑆 = ∅ → 0 ∈ ℕ0)
3		breq1 5071	. . . . . . . 8 ⊢ (𝑠 = 0 → (𝑠 < 𝑥 ↔ 0 < 𝑥))
4	3	imbi1d 344	. . . . . . 7 ⊢ (𝑠 = 0 → ((𝑠 < 𝑥 → 𝑥 ∉ 𝑆) ↔ (0 < 𝑥 → 𝑥 ∉ 𝑆)))
5	4	ralbidv 3199	. . . . . 6 ⊢ (𝑠 = 0 → (∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆) ↔ ∀𝑥 ∈ ℕ0 (0 < 𝑥 → 𝑥 ∉ 𝑆)))
6	5	adantl 484	. . . . 5 ⊢ ((𝑆 = ∅ ∧ 𝑠 = 0) → (∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆) ↔ ∀𝑥 ∈ ℕ0 (0 < 𝑥 → 𝑥 ∉ 𝑆)))
7		nnel 3134	. . . . . . . . 9 ⊢ (¬ 𝑥 ∉ 𝑆 ↔ 𝑥 ∈ 𝑆)
8		n0i 4301	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑆 → ¬ 𝑆 = ∅)
9	7, 8	sylbi 219	. . . . . . . 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 3185	. . . . 5 ⊢ (𝑆 = ∅ → ∀𝑥 ∈ ℕ0 (0 < 𝑥 → 𝑥 ∉ 𝑆))
13	2, 6, 12	rspcedvd 3628	. . . 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 10723	. . . . . . 7 ⊢ < Or ℝ
16		id 22	. . . . . . . . 9 ⊢ (𝑆 ⊆ ℕ0 → 𝑆 ⊆ ℕ0)
17		nn0ssre 11904	. . . . . . . . 9 ⊢ ℕ0 ⊆ ℝ
18	16, 17	sstrdi 3981	. . . . . . . 8 ⊢ (𝑆 ⊆ ℕ0 → 𝑆 ⊆ ℝ)
19	18	3anim3i 1150	. . . . . . 7 ⊢ ((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) → (𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℝ))
20		fisup2g 8934	. . . . . . 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 589	. . . . . 6 ⊢ ((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) → ∃𝑠 ∈ 𝑆 (∀𝑦 ∈ 𝑆 ¬ 𝑠 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑠 → ∃𝑧 ∈ 𝑆 𝑦 < 𝑧)))
22		simp3 1134	. . . . . . 7 ⊢ ((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) → 𝑆 ⊆ ℕ0)
23		breq2 5072	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑥 → (𝑠 < 𝑦 ↔ 𝑠 < 𝑥))
24	23	notbid 320	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑥 → (¬ 𝑠 < 𝑦 ↔ ¬ 𝑠 < 𝑥))
25	24	rspcva 3623	. . . . . . . . . . . . . . . . . 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 416	. . . . . . . . . . . . . . . 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 420	. . . . . . . . . . . . . 14 ⊢ (((∀𝑦 ∈ 𝑆 ¬ 𝑠 < 𝑦 ∧ ((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) ∧ 𝑠 ∈ 𝑆)) ∧ 𝑥 ∈ ℕ0) → (𝑥 ∈ 𝑆 → ¬ 𝑠 < 𝑥))
30	7, 29	syl5bi 244	. . . . . . . . . . . . 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 3184	. . . . . . . . . . 11 ⊢ ((∀𝑦 ∈ 𝑆 ¬ 𝑠 < 𝑦 ∧ ((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) ∧ 𝑠 ∈ 𝑆)) → ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆))
33	32	ex 415	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝑆 ¬ 𝑠 < 𝑦 → (((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) ∧ 𝑠 ∈ 𝑆) → ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)))
34	33	adantr 483	. . . . . . . . 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 3276	. . . . . . 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 4036	. . . . . . 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 1115	. . . 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 3102	. 2 ⊢ (𝑆 ⊆ ℕ0 → (𝑆 ∈ Fin → ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)))
43		fzfi 13343	. . . 4 ⊢ (0...𝑠) ∈ Fin
44		elfz2nn0 13001	. . . . . . . . . 10 ⊢ (𝑦 ∈ (0...𝑠) ↔ (𝑦 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0 ∧ 𝑦 ≤ 𝑠))
45	44	notbii 322	. . . . . . . . 9 ⊢ (¬ 𝑦 ∈ (0...𝑠) ↔ ¬ (𝑦 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0 ∧ 𝑦 ≤ 𝑠))
46		3ianor 1103	. . . . . . . . 9 ⊢ (¬ (𝑦 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0 ∧ 𝑦 ≤ 𝑠) ↔ (¬ 𝑦 ∈ ℕ0 ∨ ¬ 𝑠 ∈ ℕ0 ∨ ¬ 𝑦 ≤ 𝑠))
47		3orass 1086	. . . . . . . . 9 ⊢ ((¬ 𝑦 ∈ ℕ0 ∨ ¬ 𝑠 ∈ ℕ0 ∨ ¬ 𝑦 ≤ 𝑠) ↔ (¬ 𝑦 ∈ ℕ0 ∨ (¬ 𝑠 ∈ ℕ0 ∨ ¬ 𝑦 ≤ 𝑠)))
48	45, 46, 47	3bitri 299	. . . . . . . 8 ⊢ (¬ 𝑦 ∈ (0...𝑠) ↔ (¬ 𝑦 ∈ ℕ0 ∨ (¬ 𝑠 ∈ ℕ0 ∨ ¬ 𝑦 ≤ 𝑠)))
49		ssel 3963	. . . . . . . . . . . 12 ⊢ (𝑆 ⊆ ℕ0 → (𝑦 ∈ 𝑆 → 𝑦 ∈ ℕ0))
50	49	adantr 483	. . . . . . . . . . 11 ⊢ ((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) → (𝑦 ∈ 𝑆 → 𝑦 ∈ ℕ0))
51	50	adantr 483	. . . . . . . . . 10 ⊢ (((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → (𝑦 ∈ 𝑆 → 𝑦 ∈ ℕ0))
52	51	con3rr3 158	. . . . . . . . 9 ⊢ (¬ 𝑦 ∈ ℕ0 → (((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → ¬ 𝑦 ∈ 𝑆))
53		notnotb 317	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ0 ↔ ¬ ¬ 𝑦 ∈ ℕ0)
54		pm2.24 124	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ ℕ0 → (¬ 𝑠 ∈ ℕ0 → ¬ 𝑦 ∈ 𝑆))
55	54	adantl 484	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) → (¬ 𝑠 ∈ ℕ0 → ¬ 𝑦 ∈ 𝑆))
56	55	adantr 483	. . . . . . . . . . . . . 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 5072	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑦 → (𝑠 < 𝑥 ↔ 𝑠 < 𝑦))
60		neleq1 3130	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑦 → (𝑥 ∉ 𝑆 ↔ 𝑦 ∉ 𝑆))
61	59, 60	imbi12d 347	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → ((𝑠 < 𝑥 → 𝑥 ∉ 𝑆) ↔ (𝑠 < 𝑦 → 𝑦 ∉ 𝑆)))
62	61	rspcva 3623	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → (𝑠 < 𝑦 → 𝑦 ∉ 𝑆))
63		nn0re 11909	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑠 ∈ ℕ0 → 𝑠 ∈ ℝ)
64		nn0re 11909	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ ℕ0 → 𝑦 ∈ ℝ)
65		ltnle 10722	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑠 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑠 < 𝑦 ↔ ¬ 𝑦 ≤ 𝑠))
66	63, 64, 65	syl2an 597	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑠 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑠 < 𝑦 ↔ ¬ 𝑦 ≤ 𝑠))
67		df-nel 3126	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∉ 𝑆 ↔ ¬ 𝑦 ∈ 𝑆)
68	67	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑠 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑦 ∉ 𝑆 ↔ ¬ 𝑦 ∈ 𝑆))
69	66, 68	imbi12d 347	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑠 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → ((𝑠 < 𝑦 → 𝑦 ∉ 𝑆) ↔ (¬ 𝑦 ≤ 𝑠 → ¬ 𝑦 ∈ 𝑆)))
70	69	biimpd 231	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑠 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → ((𝑠 < 𝑦 → 𝑦 ∉ 𝑆) → (¬ 𝑦 ≤ 𝑠 → ¬ 𝑦 ∈ 𝑆)))
71	70	ex 415	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑠 ∈ ℕ0 → (𝑦 ∈ ℕ0 → ((𝑠 < 𝑦 → 𝑦 ∉ 𝑆) → (¬ 𝑦 ≤ 𝑠 → ¬ 𝑦 ∈ 𝑆))))
72	71	adantl 484	. . . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . 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 415	. . . . . . . . . . . . . . 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 409	. . . . . . . . . . . . 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 853	. . . . . . . . . . 11 ⊢ ((¬ 𝑠 ∈ ℕ0 ∨ ¬ 𝑦 ≤ 𝑠) → (𝑦 ∈ ℕ0 → (((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → ¬ 𝑦 ∈ 𝑆)))
81	53, 80	syl5bir 245	. . . . . . . . . 10 ⊢ ((¬ 𝑠 ∈ ℕ0 ∨ ¬ 𝑦 ≤ 𝑠) → (¬ ¬ 𝑦 ∈ ℕ0 → (((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → ¬ 𝑦 ∈ 𝑆)))
82	81	impcom 410	. . . . . . . . 9 ⊢ ((¬ ¬ 𝑦 ∈ ℕ0 ∧ (¬ 𝑠 ∈ ℕ0 ∨ ¬ 𝑦 ≤ 𝑠)) → (((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → ¬ 𝑦 ∈ 𝑆))
83	52, 82	jaoi3 1055	. . . . . . . 8 ⊢ ((¬ 𝑦 ∈ ℕ0 ∨ (¬ 𝑠 ∈ ℕ0 ∨ ¬ 𝑦 ≤ 𝑠)) → (((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → ¬ 𝑦 ∈ 𝑆))
84	48, 83	sylbi 219	. . . . . . 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 3975	. . . 4 ⊢ (((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → 𝑆 ⊆ (0...𝑠))
88		ssfi 8740	. . . 4 ⊢ (((0...𝑠) ∈ Fin ∧ 𝑆 ⊆ (0...𝑠)) → 𝑆 ∈ Fin)
89	43, 87, 88	sylancr 589	. . 3 ⊢ (((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → 𝑆 ∈ Fin)
90	89	rexlimdva2 3289	. 2 ⊢ (𝑆 ⊆ ℕ0 → (∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆) → 𝑆 ∈ Fin))
91	42, 90	impbid 214	1 ⊢ (𝑆 ⊆ ℕ0 → (𝑆 ∈ Fin ↔ ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   ∨ w3o 1082   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   ≠ wne 3018   ∉ wnel 3125  ∀wral 3140  ∃wrex 3141   ⊆ wss 3938  ∅c0 4293   class class class wbr 5068   Or wor 5475  (class class class)co 7158  Fincfn 8511  ℝcr 10538  0cc0 10539   < clt 10677   ≤ cle 10678  ℕ₀cn0 11900  ...cfz 12895

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-mulcom 10603  ax-addass 10604  ax-mulass 10605  ax-distr 10606  ax-i2m1 10607  ax-1ne0 10608  ax-1rid 10609  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612  ax-pre-lttri 10613  ax-pre-lttrn 10614  ax-pre-ltadd 10615  ax-pre-mulgt0 10616

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-1st 7691  df-2nd 7692  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-1o 8104  df-er 8291  df-en 8512  df-dom 8513  df-sdom 8514  df-fin 8515  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-sub 10874  df-neg 10875  df-nn 11641  df-n0 11901  df-z 11985  df-uz 12247  df-fz 12896

This theorem is referenced by:  rabssnn0fi  13357