Theorem ssnn0fi 13908

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 12416	. . . . . 6 ⊢ 0 ∈ ℕ0
2	1	a1i 11	. . . . 5 ⊢ (𝑆 = ∅ → 0 ∈ ℕ0)
3		breq1 5101	. . . . . . . 8 ⊢ (𝑠 = 0 → (𝑠 < 𝑥 ↔ 0 < 𝑥))
4	3	imbi1d 341	. . . . . . 7 ⊢ (𝑠 = 0 → ((𝑠 < 𝑥 → 𝑥 ∉ 𝑆) ↔ (0 < 𝑥 → 𝑥 ∉ 𝑆)))
5	4	ralbidv 3159	. . . . . 6 ⊢ (𝑠 = 0 → (∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆) ↔ ∀𝑥 ∈ ℕ0 (0 < 𝑥 → 𝑥 ∉ 𝑆)))
6	5	adantl 481	. . . . 5 ⊢ ((𝑆 = ∅ ∧ 𝑠 = 0) → (∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆) ↔ ∀𝑥 ∈ ℕ0 (0 < 𝑥 → 𝑥 ∉ 𝑆)))
7		nnel 3046	. . . . . . . . 9 ⊢ (¬ 𝑥 ∉ 𝑆 ↔ 𝑥 ∈ 𝑆)
8		n0i 4292	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑆 → ¬ 𝑆 = ∅)
9	7, 8	sylbi 217	. . . . . . . 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 3132	. . . . 5 ⊢ (𝑆 = ∅ → ∀𝑥 ∈ ℕ0 (0 < 𝑥 → 𝑥 ∉ 𝑆))
13	2, 6, 12	rspcedvd 3578	. . . 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 11213	. . . . . . 7 ⊢ < Or ℝ
16		id 22	. . . . . . . . 9 ⊢ (𝑆 ⊆ ℕ0 → 𝑆 ⊆ ℕ0)
17		nn0ssre 12405	. . . . . . . . 9 ⊢ ℕ0 ⊆ ℝ
18	16, 17	sstrdi 3946	. . . . . . . 8 ⊢ (𝑆 ⊆ ℕ0 → 𝑆 ⊆ ℝ)
19	18	3anim3i 1154	. . . . . . 7 ⊢ ((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) → (𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℝ))
20		fisup2g 9372	. . . . . . 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 1138	. . . . . . 7 ⊢ ((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) → 𝑆 ⊆ ℕ0)
23		breq2 5102	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑥 → (𝑠 < 𝑦 ↔ 𝑠 < 𝑥))
24	23	notbid 318	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑥 → (¬ 𝑠 < 𝑦 ↔ ¬ 𝑠 < 𝑥))
25	24	rspcva 3574	. . . . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . . 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 417	. . . . . . . . . . . . . 14 ⊢ (((∀𝑦 ∈ 𝑆 ¬ 𝑠 < 𝑦 ∧ ((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) ∧ 𝑠 ∈ 𝑆)) ∧ 𝑥 ∈ ℕ0) → (𝑥 ∈ 𝑆 → ¬ 𝑠 < 𝑥))
30	7, 29	biimtrid 242	. . . . . . . . . . . . 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 3128	. . . . . . . . . . 11 ⊢ ((∀𝑦 ∈ 𝑆 ¬ 𝑠 < 𝑦 ∧ ((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) ∧ 𝑠 ∈ 𝑆)) → ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆))
33	32	ex 412	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝑆 ¬ 𝑠 < 𝑦 → (((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) ∧ 𝑠 ∈ 𝑆) → ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)))
34	33	adantr 480	. . . . . . . . 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 3149	. . . . . . 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 4003	. . . . . . 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 1119	. . . 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 3015	. 2 ⊢ (𝑆 ⊆ ℕ0 → (𝑆 ∈ Fin → ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)))
43		fzfi 13895	. . . 4 ⊢ (0...𝑠) ∈ Fin
44		elfz2nn0 13534	. . . . . . . . . 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 3927	. . . . . . . . . . . 12 ⊢ (𝑆 ⊆ ℕ0 → (𝑦 ∈ 𝑆 → 𝑦 ∈ ℕ0))
50	49	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) → (𝑦 ∈ 𝑆 → 𝑦 ∈ ℕ0))
51	50	adantr 480	. . . . . . . . . 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 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) → (¬ 𝑠 ∈ ℕ0 → ¬ 𝑦 ∈ 𝑆))
56	55	adantr 480	. . . . . . . . . . . . . 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 5102	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑦 → (𝑠 < 𝑥 ↔ 𝑠 < 𝑦))
60		neleq1 3042	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑦 → (𝑥 ∉ 𝑆 ↔ 𝑦 ∉ 𝑆))
61	59, 60	imbi12d 344	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → ((𝑠 < 𝑥 → 𝑥 ∉ 𝑆) ↔ (𝑠 < 𝑦 → 𝑦 ∉ 𝑆)))
62	61	rspcva 3574	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → (𝑠 < 𝑦 → 𝑦 ∉ 𝑆))
63		nn0re 12410	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑠 ∈ ℕ0 → 𝑠 ∈ ℝ)
64		nn0re 12410	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ ℕ0 → 𝑦 ∈ ℝ)
65		ltnle 11212	. . . . . . . . . . . . . . . . . . . . . . . 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 3037	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∉ 𝑆 ↔ ¬ 𝑦 ∈ 𝑆)
68	67	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑠 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑦 ∉ 𝑆 ↔ ¬ 𝑦 ∈ 𝑆))
69	66, 68	imbi12d 344	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑠 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → ((𝑠 < 𝑦 → 𝑦 ∉ 𝑆) ↔ (¬ 𝑦 ≤ 𝑠 → ¬ 𝑦 ∈ 𝑆)))
70	69	biimpd 229	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑠 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → ((𝑠 < 𝑦 → 𝑦 ∉ 𝑆) → (¬ 𝑦 ≤ 𝑠 → ¬ 𝑦 ∈ 𝑆)))
71	70	ex 412	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑠 ∈ ℕ0 → (𝑦 ∈ ℕ0 → ((𝑠 < 𝑦 → 𝑦 ∉ 𝑆) → (¬ 𝑦 ≤ 𝑠 → ¬ 𝑦 ∈ 𝑆))))
72	71	adantl 481	. . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . 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 857	. . . . . . . . . . 11 ⊢ ((¬ 𝑠 ∈ ℕ0 ∨ ¬ 𝑦 ≤ 𝑠) → (𝑦 ∈ ℕ0 → (((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → ¬ 𝑦 ∈ 𝑆)))
81	53, 80	biimtrrid 243	. . . . . . . . . 10 ⊢ ((¬ 𝑠 ∈ ℕ0 ∨ ¬ 𝑦 ≤ 𝑠) → (¬ ¬ 𝑦 ∈ ℕ0 → (((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → ¬ 𝑦 ∈ 𝑆)))
82	81	impcom 407	. . . . . . . . 9 ⊢ ((¬ ¬ 𝑦 ∈ ℕ0 ∧ (¬ 𝑠 ∈ ℕ0 ∨ ¬ 𝑦 ≤ 𝑠)) → (((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → ¬ 𝑦 ∈ 𝑆))
83	52, 82	jaoi3 1060	. . . . . . . 8 ⊢ ((¬ 𝑦 ∈ ℕ0 ∨ (¬ 𝑠 ∈ ℕ0 ∨ ¬ 𝑦 ≤ 𝑠)) → (((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → ¬ 𝑦 ∈ 𝑆))
84	48, 83	sylbi 217	. . . . . . 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 3939	. . . 4 ⊢ (((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → 𝑆 ⊆ (0...𝑠))
88		ssfi 9097	. . . 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 3139	. 2 ⊢ (𝑆 ⊆ ℕ0 → (∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆) → 𝑆 ∈ Fin))
91	42, 90	impbid 212	1 ⊢ (𝑆 ⊆ ℕ0 → (𝑆 ∈ Fin ↔ ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∨ w3o 1085   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   ≠ wne 2932   ∉ wnel 3036  ∀wral 3051  ∃wrex 3060   ⊆ wss 3901  ∅c0 4285   class class class wbr 5098   Or wor 5531  (class class class)co 7358  Fincfn 8883  ℝcr 11025  0cc0 11026   < clt 11166   ≤ cle 11167  ℕ₀cn0 12401  ...cfz 13423

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-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-er 8635  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-n0 12402  df-z 12489  df-uz 12752  df-fz 13424

This theorem is referenced by:  rabssnn0fi  13909