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Theorem pnfnei 22477

Description: A neighborhood of +∞ contains an unbounded interval based at a real number. Together with xrtgioo 24075 (which describes neighborhoods of ℝ) and mnfnei 22478, this gives all "negative" topological information ensuring that it is not too fine (and of course iooordt 22474 and similar ensure that it has all the sets we want). (Contributed by Mario Carneiro, 3-Sep-2015.)

**Assertion**
Ref	Expression
pnfnei	⊢ ((𝐴 ∈ (ordTop‘ ≤ ) ∧ +∞ ∈ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴)

Distinct variable group: 𝑥,𝐴

Proof of Theorem pnfnei Dummy variables 𝑎 𝑏 𝑐 𝑢 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2736	. . . 4 ⊢ ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) = ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞))
2		eqid 2736	. . . 4 ⊢ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)) = ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))
3		eqid 2736	. . . 4 ⊢ ran (,) = ran (,)
4	1, 2, 3	leordtval 22470	. . 3 ⊢ (ordTop‘ ≤ ) = (topGen‘((ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ∪ ran (,)))
5	4	eleq2i 2828	. 2 ⊢ (𝐴 ∈ (ordTop‘ ≤ ) ↔ 𝐴 ∈ (topGen‘((ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ∪ ran (,))))
6		tg2 22221	. . 3 ⊢ ((𝐴 ∈ (topGen‘((ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ∪ ran (,))) ∧ +∞ ∈ 𝐴) → ∃𝑢 ∈ ((ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ∪ ran (,))(+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴))
7		elun 4095	. . . . 5 ⊢ (𝑢 ∈ ((ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ∪ ran (,)) ↔ (𝑢 ∈ (ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ∨ 𝑢 ∈ ran (,)))
8		elun 4095	. . . . . . 7 ⊢ (𝑢 ∈ (ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ↔ (𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∨ 𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))))
9		eqid 2736	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) = (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞))
10	9	elrnmpt 5897	. . . . . . . . . 10 ⊢ (𝑢 ∈ V → (𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ↔ ∃𝑦 ∈ ℝ^* 𝑢 = (𝑦(,]+∞)))
11	10	elv 3447	. . . . . . . . 9 ⊢ (𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ↔ ∃𝑦 ∈ ℝ^* 𝑢 = (𝑦(,]+∞))
12		mnfxr 11133	. . . . . . . . . . . . . 14 ⊢ -∞ ∈ ℝ^*
13	12	a1i 11	. . . . . . . . . . . . 13 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → -∞ ∈ ℝ^*)
14		simprl 768	. . . . . . . . . . . . . 14 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → 𝑦 ∈ ℝ^*)
15		0xr 11123	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ^*
16		ifcl 4518	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℝ^* ∧ 0 ∈ ℝ^) → if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ^)
17	14, 15, 16	sylancl 586	. . . . . . . . . . . . 13 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ^*)
18		pnfxr 11130	. . . . . . . . . . . . . 14 ⊢ +∞ ∈ ℝ^*
19	18	a1i 11	. . . . . . . . . . . . 13 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → +∞ ∈ ℝ^*)
20		xrmax1 13010	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → 0 ≤ if(0 ≤ 𝑦, 𝑦, 0))
21	15, 14, 20	sylancr 587	. . . . . . . . . . . . . 14 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → 0 ≤ if(0 ≤ 𝑦, 𝑦, 0))
22		ge0gtmnf 13007	. . . . . . . . . . . . . 14 ⊢ ((if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ^* ∧ 0 ≤ if(0 ≤ 𝑦, 𝑦, 0)) → -∞ < if(0 ≤ 𝑦, 𝑦, 0))
23	17, 21, 22	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → -∞ < if(0 ≤ 𝑦, 𝑦, 0))
24		simpll 764	. . . . . . . . . . . . . . . . 17 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → +∞ ∈ 𝑢)
25		simprr 770	. . . . . . . . . . . . . . . . 17 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → 𝑢 = (𝑦(,]+∞))
26	24, 25	eleqtrd 2839	. . . . . . . . . . . . . . . 16 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → +∞ ∈ (𝑦(,]+∞))
27		elioc1 13222	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℝ^* ∧ +∞ ∈ ℝ^) → (+∞ ∈ (𝑦(,]+∞) ↔ (+∞ ∈ ℝ^ ∧ 𝑦 < +∞ ∧ +∞ ≤ +∞)))
28	14, 18, 27	sylancl 586	. . . . . . . . . . . . . . . 16 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → (+∞ ∈ (𝑦(,]+∞) ↔ (+∞ ∈ ℝ^* ∧ 𝑦 < +∞ ∧ +∞ ≤ +∞)))
29	26, 28	mpbid 231	. . . . . . . . . . . . . . 15 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → (+∞ ∈ ℝ^* ∧ 𝑦 < +∞ ∧ +∞ ≤ +∞))
30	29	simp2d 1142	. . . . . . . . . . . . . 14 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → 𝑦 < +∞)
31		0ltpnf 12959	. . . . . . . . . . . . . 14 ⊢ 0 < +∞
32		breq1 5095	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = if(0 ≤ 𝑦, 𝑦, 0) → (𝑦 < +∞ ↔ if(0 ≤ 𝑦, 𝑦, 0) < +∞))
33		breq1 5095	. . . . . . . . . . . . . . 15 ⊢ (0 = if(0 ≤ 𝑦, 𝑦, 0) → (0 < +∞ ↔ if(0 ≤ 𝑦, 𝑦, 0) < +∞))
34	32, 33	ifboth 4512	. . . . . . . . . . . . . 14 ⊢ ((𝑦 < +∞ ∧ 0 < +∞) → if(0 ≤ 𝑦, 𝑦, 0) < +∞)
35	30, 31, 34	sylancl 586	. . . . . . . . . . . . 13 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → if(0 ≤ 𝑦, 𝑦, 0) < +∞)
36		xrre2 13005	. . . . . . . . . . . . 13 ⊢ (((-∞ ∈ ℝ^* ∧ if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (-∞ < if(0 ≤ 𝑦, 𝑦, 0) ∧ if(0 ≤ 𝑦, 𝑦, 0) < +∞)) → if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ)
37	13, 17, 19, 23, 35, 36	syl32anc 1377	. . . . . . . . . . . 12 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ)
38		xrmax2 13011	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → 𝑦 ≤ if(0 ≤ 𝑦, 𝑦, 0))
39	15, 14, 38	sylancr 587	. . . . . . . . . . . . . 14 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → 𝑦 ≤ if(0 ≤ 𝑦, 𝑦, 0))
40		df-ioc 13185	. . . . . . . . . . . . . . 15 ⊢ (,] = (𝑎 ∈ ℝ^, 𝑏 ∈ ℝ^ ↦ {𝑐 ∈ ℝ^* ∣ (𝑎 < 𝑐 ∧ 𝑐 ≤ 𝑏)})
41		xrlelttr 12991	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℝ^* ∧ if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ^* ∧ 𝑥 ∈ ℝ^*) → ((𝑦 ≤ if(0 ≤ 𝑦, 𝑦, 0) ∧ if(0 ≤ 𝑦, 𝑦, 0) < 𝑥) → 𝑦 < 𝑥))
42	40, 40, 41	ixxss1 13198	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℝ^* ∧ 𝑦 ≤ if(0 ≤ 𝑦, 𝑦, 0)) → (if(0 ≤ 𝑦, 𝑦, 0)(,]+∞) ⊆ (𝑦(,]+∞))
43	14, 39, 42	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → (if(0 ≤ 𝑦, 𝑦, 0)(,]+∞) ⊆ (𝑦(,]+∞))
44		simplr 766	. . . . . . . . . . . . . 14 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → 𝑢 ⊆ 𝐴)
45	25, 44	eqsstrrd 3971	. . . . . . . . . . . . 13 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → (𝑦(,]+∞) ⊆ 𝐴)
46	43, 45	sstrd 3942	. . . . . . . . . . . 12 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → (if(0 ≤ 𝑦, 𝑦, 0)(,]+∞) ⊆ 𝐴)
47		oveq1 7344	. . . . . . . . . . . . . 14 ⊢ (𝑥 = if(0 ≤ 𝑦, 𝑦, 0) → (𝑥(,]+∞) = (if(0 ≤ 𝑦, 𝑦, 0)(,]+∞))
48	47	sseq1d 3963	. . . . . . . . . . . . 13 ⊢ (𝑥 = if(0 ≤ 𝑦, 𝑦, 0) → ((𝑥(,]+∞) ⊆ 𝐴 ↔ (if(0 ≤ 𝑦, 𝑦, 0)(,]+∞) ⊆ 𝐴))
49	48	rspcev 3570	. . . . . . . . . . . 12 ⊢ ((if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ ∧ (if(0 ≤ 𝑦, 𝑦, 0)(,]+∞) ⊆ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴)
50	37, 46, 49	syl2anc 584	. . . . . . . . . . 11 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴)
51	50	rexlimdvaa 3149	. . . . . . . . . 10 ⊢ ((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → (∃𝑦 ∈ ℝ^* 𝑢 = (𝑦(,]+∞) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
52	51	com12 32	. . . . . . . . 9 ⊢ (∃𝑦 ∈ ℝ^* 𝑢 = (𝑦(,]+∞) → ((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
53	11, 52	sylbi 216	. . . . . . . 8 ⊢ (𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) → ((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
54		eqid 2736	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)) = (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))
55	54	elrnmpt 5897	. . . . . . . . . 10 ⊢ (𝑢 ∈ V → (𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)) ↔ ∃𝑦 ∈ ℝ^* 𝑢 = (-∞[,)𝑦)))
56	55	elv 3447	. . . . . . . . 9 ⊢ (𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)) ↔ ∃𝑦 ∈ ℝ^* 𝑢 = (-∞[,)𝑦))
57		pnfnlt 12965	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ^* → ¬ +∞ < 𝑦)
58		elico1 13223	. . . . . . . . . . . . . . . 16 ⊢ ((-∞ ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → (+∞ ∈ (-∞[,)𝑦) ↔ (+∞ ∈ ℝ^ ∧ -∞ ≤ +∞ ∧ +∞ < 𝑦)))
59	12, 58	mpan 687	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℝ^* → (+∞ ∈ (-∞[,)𝑦) ↔ (+∞ ∈ ℝ^* ∧ -∞ ≤ +∞ ∧ +∞ < 𝑦)))
60		simp3 1137	. . . . . . . . . . . . . . 15 ⊢ ((+∞ ∈ ℝ^* ∧ -∞ ≤ +∞ ∧ +∞ < 𝑦) → +∞ < 𝑦)
61	59, 60	syl6bi 252	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ^* → (+∞ ∈ (-∞[,)𝑦) → +∞ < 𝑦))
62	57, 61	mtod 197	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ^* → ¬ +∞ ∈ (-∞[,)𝑦))
63		eleq2 2825	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (-∞[,)𝑦) → (+∞ ∈ 𝑢 ↔ +∞ ∈ (-∞[,)𝑦)))
64	63	notbid 317	. . . . . . . . . . . . 13 ⊢ (𝑢 = (-∞[,)𝑦) → (¬ +∞ ∈ 𝑢 ↔ ¬ +∞ ∈ (-∞[,)𝑦)))
65	62, 64	syl5ibrcom 246	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ^* → (𝑢 = (-∞[,)𝑦) → ¬ +∞ ∈ 𝑢))
66	65	rexlimiv 3141	. . . . . . . . . . 11 ⊢ (∃𝑦 ∈ ℝ^* 𝑢 = (-∞[,)𝑦) → ¬ +∞ ∈ 𝑢)
67	66	pm2.21d 121	. . . . . . . . . 10 ⊢ (∃𝑦 ∈ ℝ^* 𝑢 = (-∞[,)𝑦) → (+∞ ∈ 𝑢 → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
68	67	adantrd 492	. . . . . . . . 9 ⊢ (∃𝑦 ∈ ℝ^* 𝑢 = (-∞[,)𝑦) → ((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
69	56, 68	sylbi 216	. . . . . . . 8 ⊢ (𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)) → ((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
70	53, 69	jaoi 854	. . . . . . 7 ⊢ ((𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∨ 𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) → ((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
71	8, 70	sylbi 216	. . . . . 6 ⊢ (𝑢 ∈ (ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) → ((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
72		pnfnre 11117	. . . . . . . . . 10 ⊢ +∞ ∉ ℝ
73	72	neli 3048	. . . . . . . . 9 ⊢ ¬ +∞ ∈ ℝ
74		elssuni 4885	. . . . . . . . . . 11 ⊢ (𝑢 ∈ ran (,) → 𝑢 ⊆ ∪ ran (,))
75		unirnioo 13282	. . . . . . . . . . 11 ⊢ ℝ = ∪ ran (,)
76	74, 75	sseqtrrdi 3983	. . . . . . . . . 10 ⊢ (𝑢 ∈ ran (,) → 𝑢 ⊆ ℝ)
77	76	sseld 3931	. . . . . . . . 9 ⊢ (𝑢 ∈ ran (,) → (+∞ ∈ 𝑢 → +∞ ∈ ℝ))
78	73, 77	mtoi 198	. . . . . . . 8 ⊢ (𝑢 ∈ ran (,) → ¬ +∞ ∈ 𝑢)
79	78	pm2.21d 121	. . . . . . 7 ⊢ (𝑢 ∈ ran (,) → (+∞ ∈ 𝑢 → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
80	79	adantrd 492	. . . . . 6 ⊢ (𝑢 ∈ ran (,) → ((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
81	71, 80	jaoi 854	. . . . 5 ⊢ ((𝑢 ∈ (ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ∨ 𝑢 ∈ ran (,)) → ((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
82	7, 81	sylbi 216	. . . 4 ⊢ (𝑢 ∈ ((ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ∪ ran (,)) → ((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
83	82	rexlimiv 3141	. . 3 ⊢ (∃𝑢 ∈ ((ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ∪ ran (,))(+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴)
84	6, 83	syl 17	. 2 ⊢ ((𝐴 ∈ (topGen‘((ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ∪ ran (,))) ∧ +∞ ∈ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴)
85	5, 84	sylanb 581	1 ⊢ ((𝐴 ∈ (ordTop‘ ≤ ) ∧ +∞ ∈ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 844 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ∃wrex 3070 Vcvv 3441 ∪ cun 3896 ⊆ wss 3898 ifcif 4473 ∪ cuni 4852 class class class wbr 5092 ↦ cmpt 5175 ran crn 5621 ‘cfv 6479 (class class class)co 7337 ℝcr 10971 0cc0 10972 +∞cpnf 11107 -∞cmnf 11108 ℝ^*cxr 11109 < clt 11110 ≤ cle 11111 (,)cioo 13180 (,]cioc 13181 [,)cico 13182 topGenctg 17245 ordTopcordt 17307

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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 ax-cnex 11028 ax-resscn 11029 ax-1cn 11030 ax-icn 11031 ax-addcl 11032 ax-addrcl 11033 ax-mulcl 11034 ax-mulrcl 11035 ax-mulcom 11036 ax-addass 11037 ax-mulass 11038 ax-distr 11039 ax-i2m1 11040 ax-1ne0 11041 ax-1rid 11042 ax-rnegex 11043 ax-rrecex 11044 ax-cnre 11045 ax-pre-lttri 11046 ax-pre-lttrn 11047 ax-pre-ltadd 11048 ax-pre-mulgt0 11049

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-int 4895 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-tr 5210 df-id 5518 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5575 df-we 5577 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-riota 7293 df-ov 7340 df-oprab 7341 df-mpo 7342 df-om 7781 df-1st 7899 df-2nd 7900 df-1o 8367 df-er 8569 df-en 8805 df-dom 8806 df-sdom 8807 df-fin 8808 df-fi 9268 df-pnf 11112 df-mnf 11113 df-xr 11114 df-ltxr 11115 df-le 11116 df-sub 11308 df-neg 11309 df-ioo 13184 df-ioc 13185 df-ico 13186 df-icc 13187 df-topgen 17251 df-ordt 17309 df-ps 18381 df-tsr 18382 df-top 22149 df-bases 22202

This theorem is referenced by: xrge0tsms 24103 xrlimcnp 26224 xrge0tsmsd 31604 pnfneige0 32199 xlimpnfvlem2 43722

W3C validator