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Theorem pnfnei 21547

Description: A neighborhood of +∞ contains an unbounded interval based at a real number. Together with xrtgioo 23132 (which describes neighborhoods of ℝ) and mnfnei 21548, this gives all "negative" topological information ensuring that it is not too fine (and of course iooordt 21544 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 2771	. . . 4 ⊢ ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) = ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞))
2		eqid 2771	. . . 4 ⊢ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)) = ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))
3		eqid 2771	. . . 4 ⊢ ran (,) = ran (,)
4	1, 2, 3	leordtval 21540	. . 3 ⊢ (ordTop‘ ≤ ) = (topGen‘((ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ∪ ran (,)))
5	4	eleq2i 2850	. 2 ⊢ (𝐴 ∈ (ordTop‘ ≤ ) ↔ 𝐴 ∈ (topGen‘((ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ∪ ran (,))))
6		tg2 21292	. . 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 4007	. . . . 5 ⊢ (𝑢 ∈ ((ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ∪ ran (,)) ↔ (𝑢 ∈ (ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ∨ 𝑢 ∈ ran (,)))
8		elun 4007	. . . . . . 7 ⊢ (𝑢 ∈ (ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ↔ (𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∨ 𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))))
9		eqid 2771	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) = (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞))
10	9	elrnmpt 5668	. . . . . . . . . 10 ⊢ (𝑢 ∈ V → (𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ↔ ∃𝑦 ∈ ℝ^* 𝑢 = (𝑦(,]+∞)))
11	10	elv 3413	. . . . . . . . 9 ⊢ (𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ↔ ∃𝑦 ∈ ℝ^* 𝑢 = (𝑦(,]+∞))
12		mnfxr 10496	. . . . . . . . . . . . . 14 ⊢ -∞ ∈ ℝ^*
13	12	a1i 11	. . . . . . . . . . . . 13 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → -∞ ∈ ℝ^*)
14		simprl 759	. . . . . . . . . . . . . 14 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → 𝑦 ∈ ℝ^*)
15		0xr 10485	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ^*
16		ifcl 4388	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℝ^* ∧ 0 ∈ ℝ^) → if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ^)
17	14, 15, 16	sylancl 578	. . . . . . . . . . . . 13 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ^*)
18		pnfxr 10492	. . . . . . . . . . . . . 14 ⊢ +∞ ∈ ℝ^*
19	18	a1i 11	. . . . . . . . . . . . 13 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → +∞ ∈ ℝ^*)
20		xrmax1 12383	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → 0 ≤ if(0 ≤ 𝑦, 𝑦, 0))
21	15, 14, 20	sylancr 579	. . . . . . . . . . . . . 14 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → 0 ≤ if(0 ≤ 𝑦, 𝑦, 0))
22		ge0gtmnf 12380	. . . . . . . . . . . . . 14 ⊢ ((if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ^* ∧ 0 ≤ if(0 ≤ 𝑦, 𝑦, 0)) → -∞ < if(0 ≤ 𝑦, 𝑦, 0))
23	17, 21, 22	syl2anc 576	. . . . . . . . . . . . 13 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → -∞ < if(0 ≤ 𝑦, 𝑦, 0))
24		simpll 755	. . . . . . . . . . . . . . . . 17 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → +∞ ∈ 𝑢)
25		simprr 761	. . . . . . . . . . . . . . . . 17 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → 𝑢 = (𝑦(,]+∞))
26	24, 25	eleqtrd 2861	. . . . . . . . . . . . . . . 16 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → +∞ ∈ (𝑦(,]+∞))
27		elioc1 12594	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℝ^* ∧ +∞ ∈ ℝ^) → (+∞ ∈ (𝑦(,]+∞) ↔ (+∞ ∈ ℝ^ ∧ 𝑦 < +∞ ∧ +∞ ≤ +∞)))
28	14, 18, 27	sylancl 578	. . . . . . . . . . . . . . . 16 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → (+∞ ∈ (𝑦(,]+∞) ↔ (+∞ ∈ ℝ^* ∧ 𝑦 < +∞ ∧ +∞ ≤ +∞)))
29	26, 28	mpbid 224	. . . . . . . . . . . . . . 15 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → (+∞ ∈ ℝ^* ∧ 𝑦 < +∞ ∧ +∞ ≤ +∞))
30	29	simp2d 1124	. . . . . . . . . . . . . 14 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → 𝑦 < +∞)
31		0ltpnf 12332	. . . . . . . . . . . . . 14 ⊢ 0 < +∞
32		breq1 4928	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = if(0 ≤ 𝑦, 𝑦, 0) → (𝑦 < +∞ ↔ if(0 ≤ 𝑦, 𝑦, 0) < +∞))
33		breq1 4928	. . . . . . . . . . . . . . 15 ⊢ (0 = if(0 ≤ 𝑦, 𝑦, 0) → (0 < +∞ ↔ if(0 ≤ 𝑦, 𝑦, 0) < +∞))
34	32, 33	ifboth 4382	. . . . . . . . . . . . . 14 ⊢ ((𝑦 < +∞ ∧ 0 < +∞) → if(0 ≤ 𝑦, 𝑦, 0) < +∞)
35	30, 31, 34	sylancl 578	. . . . . . . . . . . . 13 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → if(0 ≤ 𝑦, 𝑦, 0) < +∞)
36		xrre2 12378	. . . . . . . . . . . . 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 1359	. . . . . . . . . . . 12 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ)
38		xrmax2 12384	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → 𝑦 ≤ if(0 ≤ 𝑦, 𝑦, 0))
39	15, 14, 38	sylancr 579	. . . . . . . . . . . . . 14 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → 𝑦 ≤ if(0 ≤ 𝑦, 𝑦, 0))
40		df-ioc 12557	. . . . . . . . . . . . . . 15 ⊢ (,] = (𝑎 ∈ ℝ^, 𝑏 ∈ ℝ^ ↦ {𝑐 ∈ ℝ^* ∣ (𝑎 < 𝑐 ∧ 𝑐 ≤ 𝑏)})
41		xrlelttr 12364	. . . . . . . . . . . . . . 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 12570	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℝ^* ∧ 𝑦 ≤ if(0 ≤ 𝑦, 𝑦, 0)) → (if(0 ≤ 𝑦, 𝑦, 0)(,]+∞) ⊆ (𝑦(,]+∞))
43	14, 39, 42	syl2anc 576	. . . . . . . . . . . . 13 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → (if(0 ≤ 𝑦, 𝑦, 0)(,]+∞) ⊆ (𝑦(,]+∞))
44		simplr 757	. . . . . . . . . . . . . 14 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → 𝑢 ⊆ 𝐴)
45	25, 44	eqsstr3d 3889	. . . . . . . . . . . . 13 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → (𝑦(,]+∞) ⊆ 𝐴)
46	43, 45	sstrd 3861	. . . . . . . . . . . 12 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → (if(0 ≤ 𝑦, 𝑦, 0)(,]+∞) ⊆ 𝐴)
47		oveq1 6981	. . . . . . . . . . . . . 14 ⊢ (𝑥 = if(0 ≤ 𝑦, 𝑦, 0) → (𝑥(,]+∞) = (if(0 ≤ 𝑦, 𝑦, 0)(,]+∞))
48	47	sseq1d 3881	. . . . . . . . . . . . 13 ⊢ (𝑥 = if(0 ≤ 𝑦, 𝑦, 0) → ((𝑥(,]+∞) ⊆ 𝐴 ↔ (if(0 ≤ 𝑦, 𝑦, 0)(,]+∞) ⊆ 𝐴))
49	48	rspcev 3528	. . . . . . . . . . . 12 ⊢ ((if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ ∧ (if(0 ≤ 𝑦, 𝑦, 0)(,]+∞) ⊆ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴)
50	37, 46, 49	syl2anc 576	. . . . . . . . . . 11 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴)
51	50	rexlimdvaa 3223	. . . . . . . . . 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 209	. . . . . . . 8 ⊢ (𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) → ((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
54		eqid 2771	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)) = (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))
55	54	elrnmpt 5668	. . . . . . . . . 10 ⊢ (𝑢 ∈ V → (𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)) ↔ ∃𝑦 ∈ ℝ^* 𝑢 = (-∞[,)𝑦)))
56	55	elv 3413	. . . . . . . . 9 ⊢ (𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)) ↔ ∃𝑦 ∈ ℝ^* 𝑢 = (-∞[,)𝑦))
57		pnfnlt 12338	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ^* → ¬ +∞ < 𝑦)
58		elico1 12595	. . . . . . . . . . . . . . . 16 ⊢ ((-∞ ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → (+∞ ∈ (-∞[,)𝑦) ↔ (+∞ ∈ ℝ^ ∧ -∞ ≤ +∞ ∧ +∞ < 𝑦)))
59	12, 58	mpan 678	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℝ^* → (+∞ ∈ (-∞[,)𝑦) ↔ (+∞ ∈ ℝ^* ∧ -∞ ≤ +∞ ∧ +∞ < 𝑦)))
60		simp3 1119	. . . . . . . . . . . . . . 15 ⊢ ((+∞ ∈ ℝ^* ∧ -∞ ≤ +∞ ∧ +∞ < 𝑦) → +∞ < 𝑦)
61	59, 60	syl6bi 245	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ^* → (+∞ ∈ (-∞[,)𝑦) → +∞ < 𝑦))
62	57, 61	mtod 190	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ^* → ¬ +∞ ∈ (-∞[,)𝑦))
63		eleq2 2847	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (-∞[,)𝑦) → (+∞ ∈ 𝑢 ↔ +∞ ∈ (-∞[,)𝑦)))
64	63	notbid 310	. . . . . . . . . . . . 13 ⊢ (𝑢 = (-∞[,)𝑦) → (¬ +∞ ∈ 𝑢 ↔ ¬ +∞ ∈ (-∞[,)𝑦)))
65	62, 64	syl5ibrcom 239	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ^* → (𝑢 = (-∞[,)𝑦) → ¬ +∞ ∈ 𝑢))
66	65	rexlimiv 3218	. . . . . . . . . . 11 ⊢ (∃𝑦 ∈ ℝ^* 𝑢 = (-∞[,)𝑦) → ¬ +∞ ∈ 𝑢)
67	66	pm2.21d 119	. . . . . . . . . 10 ⊢ (∃𝑦 ∈ ℝ^* 𝑢 = (-∞[,)𝑦) → (+∞ ∈ 𝑢 → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
68	67	adantrd 484	. . . . . . . . 9 ⊢ (∃𝑦 ∈ ℝ^* 𝑢 = (-∞[,)𝑦) → ((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
69	56, 68	sylbi 209	. . . . . . . 8 ⊢ (𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)) → ((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
70	53, 69	jaoi 844	. . . . . . 7 ⊢ ((𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∨ 𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) → ((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
71	8, 70	sylbi 209	. . . . . 6 ⊢ (𝑢 ∈ (ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) → ((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
72		pnfnre 10479	. . . . . . . . . 10 ⊢ +∞ ∉ ℝ
73	72	neli 3068	. . . . . . . . 9 ⊢ ¬ +∞ ∈ ℝ
74		elssuni 4737	. . . . . . . . . . 11 ⊢ (𝑢 ∈ ran (,) → 𝑢 ⊆ ∪ ran (,))
75		unirnioo 12651	. . . . . . . . . . 11 ⊢ ℝ = ∪ ran (,)
76	74, 75	syl6sseqr 3901	. . . . . . . . . 10 ⊢ (𝑢 ∈ ran (,) → 𝑢 ⊆ ℝ)
77	76	sseld 3850	. . . . . . . . 9 ⊢ (𝑢 ∈ ran (,) → (+∞ ∈ 𝑢 → +∞ ∈ ℝ))
78	73, 77	mtoi 191	. . . . . . . 8 ⊢ (𝑢 ∈ ran (,) → ¬ +∞ ∈ 𝑢)
79	78	pm2.21d 119	. . . . . . 7 ⊢ (𝑢 ∈ ran (,) → (+∞ ∈ 𝑢 → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
80	79	adantrd 484	. . . . . 6 ⊢ (𝑢 ∈ ran (,) → ((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
81	71, 80	jaoi 844	. . . . 5 ⊢ ((𝑢 ∈ (ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ∨ 𝑢 ∈ ran (,)) → ((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
82	7, 81	sylbi 209	. . . 4 ⊢ (𝑢 ∈ ((ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ∪ ran (,)) → ((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
83	82	rexlimiv 3218	. . 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 573	1 ⊢ ((𝐴 ∈ (ordTop‘ ≤ ) ∧ +∞ ∈ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 387 ∨ wo 834 ∧ w3a 1069 = wceq 1508 ∈ wcel 2051 ∃wrex 3082 Vcvv 3408 ∪ cun 3820 ⊆ wss 3822 ifcif 4344 ∪ cuni 4708 class class class wbr 4925 ↦ cmpt 5004 ran crn 5404 ‘cfv 6185 (class class class)co 6974 ℝcr 10332 0cc0 10333 +∞cpnf 10469 -∞cmnf 10470 ℝ^*cxr 10471 < clt 10472 ≤ cle 10473 (,)cioo 12552 (,]cioc 12553 [,)cico 12554 topGenctg 16565 ordTopcordt 16626

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410

This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-nel 3067 df-ral 3086 df-rex 3087 df-reu 3088 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-int 4746 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-om 7395 df-1st 7499 df-2nd 7500 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-1o 7903 df-oadd 7907 df-er 8087 df-en 8305 df-dom 8306 df-sdom 8307 df-fin 8308 df-fi 8668 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-ioo 12556 df-ioc 12557 df-ico 12558 df-icc 12559 df-topgen 16571 df-ordt 16628 df-ps 17680 df-tsr 17681 df-top 21221 df-bases 21273

This theorem is referenced by: xrge0tsms 23160 xrlimcnp 25263 xrge0tsmsd 30562 pnfneige0 30870 xlimpnfvlem2 41583

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