Theorem limsupre2lem 46082

Description: Given a function on the extended reals, its supremum limit is real if and only if two condition holds: 1. there is a real number that is smaller than the function, at some point, in any upper part of the reals; 2. there is a real number that is eventually larger than the function. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
limsupre2lem.1	⊢ Ⅎ𝑗𝐹
limsupre2lem.2	⊢ (𝜑 → 𝐴 ⊆ ℝ)
limsupre2lem.3	⊢ (𝜑 → 𝐹:𝐴⟶ℝ*)

Assertion

Ref	Expression
limsupre2lem	⊢ (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 < (𝐹‘𝑗)) ∧ ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑥))))

Distinct variable groups:   𝐴,𝑗,𝑘,𝑥   𝑘,𝐹,𝑥   𝜑,𝑗,𝑘,𝑥

Allowed substitution hint:   𝐹(𝑗)

Proof of Theorem limsupre2lem

Step	Hyp	Ref	Expression
1		limsupre2lem.3	. . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*)
2		reex 11129	. . . . . . 7 ⊢ ℝ ∈ V
3	2	a1i 11	. . . . . 6 ⊢ (𝜑 → ℝ ∈ V)
4		limsupre2lem.2	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
5	3, 4	ssexd 5271	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ V)
6	1, 5	fexd 7183	. . . 4 ⊢ (𝜑 → 𝐹 ∈ V)
7	6	limsupcld 46048	. . 3 ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ*)
8		xrre4 45769	. . 3 ⊢ ((lim sup‘𝐹) ∈ ℝ* → ((lim sup‘𝐹) ∈ ℝ ↔ ((lim sup‘𝐹) ≠ -∞ ∧ (lim sup‘𝐹) ≠ +∞)))
9	7, 8	syl 17	. 2 ⊢ (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ ((lim sup‘𝐹) ≠ -∞ ∧ (lim sup‘𝐹) ≠ +∞)))
10		df-ne 2934	. . . . 5 ⊢ ((lim sup‘𝐹) ≠ -∞ ↔ ¬ (lim sup‘𝐹) = -∞)
11	10	a1i 11	. . . 4 ⊢ (𝜑 → ((lim sup‘𝐹) ≠ -∞ ↔ ¬ (lim sup‘𝐹) = -∞))
12		limsupre2lem.1	. . . . . 6 ⊢ Ⅎ𝑗𝐹
13	12, 4, 1	limsupmnf 46079	. . . . 5 ⊢ (𝜑 → ((lim sup‘𝐹) = -∞ ↔ ∀𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)))
14	13	notbid 318	. . . 4 ⊢ (𝜑 → (¬ (lim sup‘𝐹) = -∞ ↔ ¬ ∀𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)))
15		annim 403	. . . . . . . . . . . 12 ⊢ ((𝑘 ≤ 𝑗 ∧ ¬ (𝐹‘𝑗) ≤ 𝑥) ↔ ¬ (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))
16	15	rexbii 3085	. . . . . . . . . . 11 ⊢ (∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ ¬ (𝐹‘𝑗) ≤ 𝑥) ↔ ∃𝑗 ∈ 𝐴 ¬ (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))
17		rexnal 3090	. . . . . . . . . . 11 ⊢ (∃𝑗 ∈ 𝐴 ¬ (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) ↔ ¬ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))
18	16, 17	bitri 275	. . . . . . . . . 10 ⊢ (∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ ¬ (𝐹‘𝑗) ≤ 𝑥) ↔ ¬ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))
19	18	ralbii 3084	. . . . . . . . 9 ⊢ (∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ ¬ (𝐹‘𝑗) ≤ 𝑥) ↔ ∀𝑘 ∈ ℝ ¬ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))
20		ralnex 3064	. . . . . . . . 9 ⊢ (∀𝑘 ∈ ℝ ¬ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) ↔ ¬ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))
21	19, 20	bitri 275	. . . . . . . 8 ⊢ (∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ ¬ (𝐹‘𝑗) ≤ 𝑥) ↔ ¬ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))
22	21	rexbii 3085	. . . . . . 7 ⊢ (∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ ¬ (𝐹‘𝑗) ≤ 𝑥) ↔ ∃𝑥 ∈ ℝ ¬ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))
23		rexnal 3090	. . . . . . 7 ⊢ (∃𝑥 ∈ ℝ ¬ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) ↔ ¬ ∀𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))
24	22, 23	bitr2i 276	. . . . . 6 ⊢ (¬ ∀𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ ¬ (𝐹‘𝑗) ≤ 𝑥))
25	24	a1i 11	. . . . 5 ⊢ (𝜑 → (¬ ∀𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ ¬ (𝐹‘𝑗) ≤ 𝑥)))
26		simplr 769	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) → 𝑥 ∈ ℝ)
27	26	rexrd 11194	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) → 𝑥 ∈ ℝ*)
28	1	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝐹:𝐴⟶ℝ*)
29	28	ffvelcdmda 7038	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) → (𝐹‘𝑗) ∈ ℝ*)
30	27, 29	xrltnled 11212	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) → (𝑥 < (𝐹‘𝑗) ↔ ¬ (𝐹‘𝑗) ≤ 𝑥))
31	30	bicomd 223	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) → (¬ (𝐹‘𝑗) ≤ 𝑥 ↔ 𝑥 < (𝐹‘𝑗)))
32	31	anbi2d 631	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) → ((𝑘 ≤ 𝑗 ∧ ¬ (𝐹‘𝑗) ≤ 𝑥) ↔ (𝑘 ≤ 𝑗 ∧ 𝑥 < (𝐹‘𝑗))))
33	32	rexbidva 3160	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ ¬ (𝐹‘𝑗) ≤ 𝑥) ↔ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 < (𝐹‘𝑗))))
34	33	ralbidv 3161	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ ¬ (𝐹‘𝑗) ≤ 𝑥) ↔ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 < (𝐹‘𝑗))))
35	34	rexbidva 3160	. . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ ¬ (𝐹‘𝑗) ≤ 𝑥) ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 < (𝐹‘𝑗))))
36	25, 35	bitrd 279	. . . 4 ⊢ (𝜑 → (¬ ∀𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 < (𝐹‘𝑗))))
37	11, 14, 36	3bitrd 305	. . 3 ⊢ (𝜑 → ((lim sup‘𝐹) ≠ -∞ ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 < (𝐹‘𝑗))))
38		df-ne 2934	. . . . 5 ⊢ ((lim sup‘𝐹) ≠ +∞ ↔ ¬ (lim sup‘𝐹) = +∞)
39	38	a1i 11	. . . 4 ⊢ (𝜑 → ((lim sup‘𝐹) ≠ +∞ ↔ ¬ (lim sup‘𝐹) = +∞))
40	12, 4, 1	limsuppnf 46069	. . . . 5 ⊢ (𝜑 → ((lim sup‘𝐹) = +∞ ↔ ∀𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))))
41	40	notbid 318	. . . 4 ⊢ (𝜑 → (¬ (lim sup‘𝐹) = +∞ ↔ ¬ ∀𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))))
42	29, 27	xrltnled 11212	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) → ((𝐹‘𝑗) < 𝑥 ↔ ¬ 𝑥 ≤ (𝐹‘𝑗)))
43	42	imbi2d 340	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) → ((𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑥) ↔ (𝑘 ≤ 𝑗 → ¬ 𝑥 ≤ (𝐹‘𝑗))))
44	43	ralbidva 3159	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑥) ↔ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → ¬ 𝑥 ≤ (𝐹‘𝑗))))
45	44	rexbidv 3162	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑥) ↔ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → ¬ 𝑥 ≤ (𝐹‘𝑗))))
46	45	rexbidva 3160	. . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑥) ↔ ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → ¬ 𝑥 ≤ (𝐹‘𝑗))))
47		imnan 399	. . . . . . . . . . . 12 ⊢ ((𝑘 ≤ 𝑗 → ¬ 𝑥 ≤ (𝐹‘𝑗)) ↔ ¬ (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))
48	47	ralbii 3084	. . . . . . . . . . 11 ⊢ (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → ¬ 𝑥 ≤ (𝐹‘𝑗)) ↔ ∀𝑗 ∈ 𝐴 ¬ (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))
49		ralnex 3064	. . . . . . . . . . 11 ⊢ (∀𝑗 ∈ 𝐴 ¬ (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ↔ ¬ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))
50	48, 49	bitri 275	. . . . . . . . . 10 ⊢ (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → ¬ 𝑥 ≤ (𝐹‘𝑗)) ↔ ¬ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))
51	50	rexbii 3085	. . . . . . . . 9 ⊢ (∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → ¬ 𝑥 ≤ (𝐹‘𝑗)) ↔ ∃𝑘 ∈ ℝ ¬ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))
52		rexnal 3090	. . . . . . . . 9 ⊢ (∃𝑘 ∈ ℝ ¬ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ↔ ¬ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))
53	51, 52	bitri 275	. . . . . . . 8 ⊢ (∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → ¬ 𝑥 ≤ (𝐹‘𝑗)) ↔ ¬ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))
54	53	rexbii 3085	. . . . . . 7 ⊢ (∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → ¬ 𝑥 ≤ (𝐹‘𝑗)) ↔ ∃𝑥 ∈ ℝ ¬ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))
55		rexnal 3090	. . . . . . 7 ⊢ (∃𝑥 ∈ ℝ ¬ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ↔ ¬ ∀𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))
56	54, 55	bitri 275	. . . . . 6 ⊢ (∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → ¬ 𝑥 ≤ (𝐹‘𝑗)) ↔ ¬ ∀𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))
57	56	a1i 11	. . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → ¬ 𝑥 ≤ (𝐹‘𝑗)) ↔ ¬ ∀𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))))
58	46, 57	bitr2d 280	. . . 4 ⊢ (𝜑 → (¬ ∀𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ↔ ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑥)))
59	39, 41, 58	3bitrd 305	. . 3 ⊢ (𝜑 → ((lim sup‘𝐹) ≠ +∞ ↔ ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑥)))
60	37, 59	anbi12d 633	. 2 ⊢ (𝜑 → (((lim sup‘𝐹) ≠ -∞ ∧ (lim sup‘𝐹) ≠ +∞) ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 < (𝐹‘𝑗)) ∧ ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑥))))
61	9, 60	bitrd 279	1 ⊢ (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 < (𝐹‘𝑗)) ∧ ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑥))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Ⅎwnfc 2884   ≠ wne 2933  ∀wral 3052  ∃wrex 3062  Vcvv 3442   ⊆ wss 3903   class class class wbr 5100  ⟶wf 6496  ‘cfv 6500  ℝcr 11037  +∞cpnf 11175  -∞cmnf 11176  ℝ^*cxr 11177   < clt 11178   ≤ cle 11179  lim supclsp 15405

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-pre-sup 11116

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-po 5540  df-so 5541  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-er 8645  df-en 8896  df-dom 8897  df-sdom 8898  df-sup 9357  df-inf 9358  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-ico 13279  df-limsup 15406

This theorem is referenced by:  limsupre2  46083