Theorem ssnn0fi 14128

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 12621	. . . . . 6 ⊢ 0 ∈ ℕ0
2	1	a1i 11	. . . . 5 ⊢ (𝑆 = ∅ → 0 ∈ ℕ0)
3		breq1 5106	. . . . . . . 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 3186	. . . . . 6 ⊢ (𝑠 = 0 → (∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆) ↔ ∀𝑥 ∈ ℕ0 (0 < 𝑥 → 𝑥 ∉ 𝑆)))
6	5	adantl 487	. . . . 5 ⊢ ((𝑆 = ∅ ∧ 𝑠 = 0) → (∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆) ↔ ∀𝑥 ∈ ℕ0 (0 < 𝑥 → 𝑥 ∉ 𝑆)))
7		nnel 3072	. . . . . . . . 9 ⊢ (¬ 𝑥 ∉ 𝑆 ↔ 𝑥 ∈ 𝑆)
8		n0i 4286	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑆 → ¬ 𝑆 = ∅)
9	7, 8	sylbi 220	. . . . . . . 8 ⊢ (¬ 𝑥 ∉ 𝑆 → ¬ 𝑆 = ∅)
10	9	con4i 115	. . . . . . 7 ⊢ (𝑆 = ∅ → 𝑥 ∉ 𝑆)
11	10	a1d 26	. . . . . 6 ⊢ (𝑆 = ∅ → (0 < 𝑥 → 𝑥 ∉ 𝑆))
12	11	ralrimivw 3159	. . . . 5 ⊢ (𝑆 = ∅ → ∀𝑥 ∈ ℕ0 (0 < 𝑥 → 𝑥 ∉ 𝑆))
13	2, 6, 12	rspcedvd 3579	. . . 4 ⊢ (𝑆 = ∅ → ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆))
14	13	2a1d 27	. . 3 ⊢ (𝑆 = ∅ → (𝑆 ⊆ ℕ0 → (𝑆 ∈ Fin → ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆))))
15		ltso 11390	. . . . . . 7 ⊢ < Or ℝ
16		id 23	. . . . . . . . 9 ⊢ (𝑆 ⊆ ℕ0 → 𝑆 ⊆ ℕ0)
17		nn0ssre 12610	. . . . . . . . 9 ⊢ ℕ0 ⊆ ℝ
18	16, 17	sstrdi 3943	. . . . . . . 8 ⊢ (𝑆 ⊆ ℕ0 → 𝑆 ⊆ ℝ)
19	18	3anim3i 1172	. . . . . . 7 ⊢ ((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) → (𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℝ))
20		fisup2g 9461	. . . . . . 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 599	. . . . . 6 ⊢ ((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) → ∃𝑠 ∈ 𝑆 (∀𝑦 ∈ 𝑆 ¬ 𝑠 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑠 → ∃𝑧 ∈ 𝑆 𝑦 < 𝑧)))
22		simp3 1156	. . . . . . 7 ⊢ ((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) → 𝑆 ⊆ ℕ0)
23		breq2 5107	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑥 → (𝑠 < 𝑦 ↔ 𝑠 < 𝑥))
24	23	notbid 321	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑥 → (¬ 𝑠 < 𝑦 ↔ ¬ 𝑠 < 𝑥))
25	24	rspcva 3575	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 ¬ 𝑠 < 𝑦) → ¬ 𝑠 < 𝑥)
26	25	2a1d 27	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 ¬ 𝑠 < 𝑦) → (𝑥 ∈ ℕ0 → (((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) ∧ 𝑠 ∈ 𝑆) → ¬ 𝑠 < 𝑥)))
27	26	expcom 419	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑦 ∈ 𝑆 ¬ 𝑠 < 𝑦 → (𝑥 ∈ 𝑆 → (𝑥 ∈ ℕ0 → (((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) ∧ 𝑠 ∈ 𝑆) → ¬ 𝑠 < 𝑥))))
28	27	com24 96	. . . . . . . . . . . . . . 15 ⊢ (∀𝑦 ∈ 𝑆 ¬ 𝑠 < 𝑦 → (((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) ∧ 𝑠 ∈ 𝑆) → (𝑥 ∈ ℕ0 → (𝑥 ∈ 𝑆 → ¬ 𝑠 < 𝑥))))
29	28	imp31 423	. . . . . . . . . . . . . 14 ⊢ (((∀𝑦 ∈ 𝑆 ¬ 𝑠 < 𝑦 ∧ ((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) ∧ 𝑠 ∈ 𝑆)) ∧ 𝑥 ∈ ℕ0) → (𝑥 ∈ 𝑆 → ¬ 𝑠 < 𝑥))
30	7, 29	biimtrid 245	. . . . . . . . . . . . 13 ⊢ (((∀𝑦 ∈ 𝑆 ¬ 𝑠 < 𝑦 ∧ ((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) ∧ 𝑠 ∈ 𝑆)) ∧ 𝑥 ∈ ℕ0) → (¬ 𝑥 ∉ 𝑆 → ¬ 𝑠 < 𝑥))
31	30	con4d 116	. . . . . . . . . . . 12 ⊢ (((∀𝑦 ∈ 𝑆 ¬ 𝑠 < 𝑦 ∧ ((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) ∧ 𝑠 ∈ 𝑆)) ∧ 𝑥 ∈ ℕ0) → (𝑠 < 𝑥 → 𝑥 ∉ 𝑆))
32	31	ralrimiva 3155	. . . . . . . . . . 11 ⊢ ((∀𝑦 ∈ 𝑆 ¬ 𝑠 < 𝑦 ∧ ((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) ∧ 𝑠 ∈ 𝑆)) → ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆))
33	32	ex 418	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝑆 ¬ 𝑠 < 𝑦 → (((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) ∧ 𝑠 ∈ 𝑆) → ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)))
34	33	adantr 486	. . . . . . . . 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 33	. . . . . . . 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 3176	. . . . . . 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 4001	. . . . . . 7 ⊢ (𝑆 ⊆ ℕ0 → (∃𝑠 ∈ 𝑆 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆) → ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)))
38	22, 36, 37	sylsyld 62	. . . . . 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 16	. . . . 5 ⊢ ((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) → ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆))
40	39	3exp 1137	. . . 4 ⊢ (𝑆 ∈ Fin → (𝑆 ≠ ∅ → (𝑆 ⊆ ℕ0 → ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆))))
41	40	com3l 90	. . 3 ⊢ (𝑆 ≠ ∅ → (𝑆 ⊆ ℕ0 → (𝑆 ∈ Fin → ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆))))
42	14, 41	pm2.61ine 3039	. 2 ⊢ (𝑆 ⊆ ℕ0 → (𝑆 ∈ Fin → ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)))
43		fzfi 14115	. . . 4 ⊢ (0...𝑠) ∈ Fin
44		elfz2nn0 13752	. . . . . . . . . 10 ⊢ (𝑦 ∈ (0...𝑠) ↔ (𝑦 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0 ∧ 𝑦 ≤ 𝑠))
45	44	notbii 323	. . . . . . . . 9 ⊢ (¬ 𝑦 ∈ (0...𝑠) ↔ ¬ (𝑦 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0 ∧ 𝑦 ≤ 𝑠))
46		3ianor 1124	. . . . . . . . 9 ⊢ (¬ (𝑦 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0 ∧ 𝑦 ≤ 𝑠) ↔ (¬ 𝑦 ∈ ℕ0 ∨ ¬ 𝑠 ∈ ℕ0 ∨ ¬ 𝑦 ≤ 𝑠))
47		3orass 1106	. . . . . . . . 9 ⊢ ((¬ 𝑦 ∈ ℕ0 ∨ ¬ 𝑠 ∈ ℕ0 ∨ ¬ 𝑦 ≤ 𝑠) ↔ (¬ 𝑦 ∈ ℕ0 ∨ (¬ 𝑠 ∈ ℕ0 ∨ ¬ 𝑦 ≤ 𝑠)))
48	45, 46, 47	3bitri 300	. . . . . . . 8 ⊢ (¬ 𝑦 ∈ (0...𝑠) ↔ (¬ 𝑦 ∈ ℕ0 ∨ (¬ 𝑠 ∈ ℕ0 ∨ ¬ 𝑦 ≤ 𝑠)))
49		ssel 3925	. . . . . . . . . . . 12 ⊢ (𝑆 ⊆ ℕ0 → (𝑦 ∈ 𝑆 → 𝑦 ∈ ℕ0))
50	49	adantr 486	. . . . . . . . . . 11 ⊢ ((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) → (𝑦 ∈ 𝑆 → 𝑦 ∈ ℕ0))
51	50	adantr 486	. . . . . . . . . 10 ⊢ (((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → (𝑦 ∈ 𝑆 → 𝑦 ∈ ℕ0))
52	51	con3rr3 156	. . . . . . . . 9 ⊢ (¬ 𝑦 ∈ ℕ0 → (((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → ¬ 𝑦 ∈ 𝑆))
53		notnotb 318	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ0 ↔ ¬ ¬ 𝑦 ∈ ℕ0)
54		pm2.24 125	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ ℕ0 → (¬ 𝑠 ∈ ℕ0 → ¬ 𝑦 ∈ 𝑆))
55	54	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) → (¬ 𝑠 ∈ ℕ0 → ¬ 𝑦 ∈ 𝑆))
56	55	adantr 486	. . . . . . . . . . . . . 14 ⊢ (((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → (¬ 𝑠 ∈ ℕ0 → ¬ 𝑦 ∈ 𝑆))
57	56	com12 33	. . . . . . . . . . . . 13 ⊢ (¬ 𝑠 ∈ ℕ0 → (((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → ¬ 𝑦 ∈ 𝑆))
58	57	a1d 26	. . . . . . . . . . . 12 ⊢ (¬ 𝑠 ∈ ℕ0 → (𝑦 ∈ ℕ0 → (((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → ¬ 𝑦 ∈ 𝑆)))
59		breq2 5107	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑦 → (𝑠 < 𝑥 ↔ 𝑠 < 𝑦))
60		neleq1 3068	. . . . . . . . . . . . . . . . . . 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 3575	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → (𝑠 < 𝑦 → 𝑦 ∉ 𝑆))
63		nn0re 12615	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑠 ∈ ℕ0 → 𝑠 ∈ ℝ)
64		nn0re 12615	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ ℕ0 → 𝑦 ∈ ℝ)
65		ltnle 11389	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑠 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑠 < 𝑦 ↔ ¬ 𝑦 ≤ 𝑠))
66	63, 64, 65	syl2an 608	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑠 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑠 < 𝑦 ↔ ¬ 𝑦 ≤ 𝑠))
67		df-nel 3063	. . . . . . . . . . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑠 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → ((𝑠 < 𝑦 → 𝑦 ∉ 𝑆) → (¬ 𝑦 ≤ 𝑠 → ¬ 𝑦 ∈ 𝑆)))
71	70	ex 418	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑠 ∈ ℕ0 → (𝑦 ∈ ℕ0 → ((𝑠 < 𝑦 → 𝑦 ∉ 𝑆) → (¬ 𝑦 ≤ 𝑠 → ¬ 𝑦 ∈ 𝑆))))
72	71	adantl 487	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) → (𝑦 ∈ ℕ0 → ((𝑠 < 𝑦 → 𝑦 ∉ 𝑆) → (¬ 𝑦 ≤ 𝑠 → ¬ 𝑦 ∈ 𝑆))))
73	72	com12 33	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ℕ0 → ((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) → ((𝑠 < 𝑦 → 𝑦 ∉ 𝑆) → (¬ 𝑦 ≤ 𝑠 → ¬ 𝑦 ∈ 𝑆))))
74	73	adantr 486	. . . . . . . . . . . . . . . . 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 45	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → ((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) → (¬ 𝑦 ≤ 𝑠 → ¬ 𝑦 ∈ 𝑆)))
76	75	ex 418	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℕ0 → (∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆) → ((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) → (¬ 𝑦 ≤ 𝑠 → ¬ 𝑦 ∈ 𝑆))))
77	76	com13 89	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) → (∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆) → (𝑦 ∈ ℕ0 → (¬ 𝑦 ≤ 𝑠 → ¬ 𝑦 ∈ 𝑆))))
78	77	imp 412	. . . . . . . . . . . . 13 ⊢ (((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → (𝑦 ∈ ℕ0 → (¬ 𝑦 ≤ 𝑠 → ¬ 𝑦 ∈ 𝑆)))
79	78	com13 89	. . . . . . . . . . . 12 ⊢ (¬ 𝑦 ≤ 𝑠 → (𝑦 ∈ ℕ0 → (((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → ¬ 𝑦 ∈ 𝑆)))
80	58, 79	jaoi 871	. . . . . . . . . . 11 ⊢ ((¬ 𝑠 ∈ ℕ0 ∨ ¬ 𝑦 ≤ 𝑠) → (𝑦 ∈ ℕ0 → (((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → ¬ 𝑦 ∈ 𝑆)))
81	53, 80	biimtrrid 246	. . . . . . . . . 10 ⊢ ((¬ 𝑠 ∈ ℕ0 ∨ ¬ 𝑦 ≤ 𝑠) → (¬ ¬ 𝑦 ∈ ℕ0 → (((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → ¬ 𝑦 ∈ 𝑆)))
82	81	impcom 413	. . . . . . . . 9 ⊢ ((¬ ¬ 𝑦 ∈ ℕ0 ∧ (¬ 𝑠 ∈ ℕ0 ∨ ¬ 𝑦 ≤ 𝑠)) → (((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → ¬ 𝑦 ∈ 𝑆))
83	52, 82	jaoi3 1076	. . . . . . . 8 ⊢ ((¬ 𝑦 ∈ ℕ0 ∨ (¬ 𝑠 ∈ ℕ0 ∨ ¬ 𝑦 ≤ 𝑠)) → (((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → ¬ 𝑦 ∈ 𝑆))
84	48, 83	sylbi 220	. . . . . . 7 ⊢ (¬ 𝑦 ∈ (0...𝑠) → (((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → ¬ 𝑦 ∈ 𝑆))
85	84	com12 33	. . . . . 6 ⊢ (((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → (¬ 𝑦 ∈ (0...𝑠) → ¬ 𝑦 ∈ 𝑆))
86	85	con4d 116	. . . . 5 ⊢ (((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → (𝑦 ∈ 𝑆 → 𝑦 ∈ (0...𝑠)))
87	86	ssrdv 3937	. . . 4 ⊢ (((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → 𝑆 ⊆ (0...𝑠))
88		ssfi 9188	. . . 4 ⊢ (((0...𝑠) ∈ Fin ∧ 𝑆 ⊆ (0...𝑠)) → 𝑆 ∈ Fin)
89	43, 87, 88	sylancr 599	. . 3 ⊢ (((𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)) → 𝑆 ∈ Fin)
90	89	rexlimdva2 3166	. 2 ⊢ (𝑆 ⊆ ℕ0 → (∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆) → 𝑆 ∈ Fin))
91	42, 90	impbid 215	1 ⊢ (𝑆 ⊆ ℕ0 → (𝑆 ∈ Fin ↔ ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∨ w3o 1102   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2956   ∉ wnel 3062  ∀wral 3077  ∃wrex 3087   ⊆ wss 3899  ∅c0 4279   class class class wbr 5103   Or wor 5558  (class class class)co 7420  Fincfn 8973  ℝcr 11199  0cc0 11200   < clt 11343   ≤ cle 11344  ℕ₀cn0 12606  ...cfz 13639

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-er 8717  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-n0 12607  df-z 12694  df-uz 12966  df-fz 13640

This theorem is used by:  rabssnn0fi  14129