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Theorem pnfnei 22071

Description: A neighborhood of +∞ contains an unbounded interval based at a real number. Together with xrtgioo 23657 (which describes neighborhoods of ℝ) and mnfnei 22072, this gives all "negative" topological information ensuring that it is not too fine (and of course iooordt 22068 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 22064	. . 3 ⊢ (ordTop‘ ≤ ) = (topGen‘((ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ∪ ran (,)))
5	4	eleq2i 2822	. 2 ⊢ (𝐴 ∈ (ordTop‘ ≤ ) ↔ 𝐴 ∈ (topGen‘((ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ∪ ran (,))))
6		tg2 21816	. . 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 4049	. . . . 5 ⊢ (𝑢 ∈ ((ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ∪ ran (,)) ↔ (𝑢 ∈ (ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ∨ 𝑢 ∈ ran (,)))
8		elun 4049	. . . . . . 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 5810	. . . . . . . . . 10 ⊢ (𝑢 ∈ V → (𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ↔ ∃𝑦 ∈ ℝ^* 𝑢 = (𝑦(,]+∞)))
11	10	elv 3404	. . . . . . . . 9 ⊢ (𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ↔ ∃𝑦 ∈ ℝ^* 𝑢 = (𝑦(,]+∞))
12		mnfxr 10855	. . . . . . . . . . . . . 14 ⊢ -∞ ∈ ℝ^*
13	12	a1i 11	. . . . . . . . . . . . 13 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → -∞ ∈ ℝ^*)
14		simprl 771	. . . . . . . . . . . . . 14 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → 𝑦 ∈ ℝ^*)
15		0xr 10845	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ^*
16		ifcl 4470	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℝ^* ∧ 0 ∈ ℝ^) → if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ^)
17	14, 15, 16	sylancl 589	. . . . . . . . . . . . 13 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ^*)
18		pnfxr 10852	. . . . . . . . . . . . . 14 ⊢ +∞ ∈ ℝ^*
19	18	a1i 11	. . . . . . . . . . . . 13 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → +∞ ∈ ℝ^*)
20		xrmax1 12730	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → 0 ≤ if(0 ≤ 𝑦, 𝑦, 0))
21	15, 14, 20	sylancr 590	. . . . . . . . . . . . . 14 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → 0 ≤ if(0 ≤ 𝑦, 𝑦, 0))
22		ge0gtmnf 12727	. . . . . . . . . . . . . 14 ⊢ ((if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ^* ∧ 0 ≤ if(0 ≤ 𝑦, 𝑦, 0)) → -∞ < if(0 ≤ 𝑦, 𝑦, 0))
23	17, 21, 22	syl2anc 587	. . . . . . . . . . . . 13 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → -∞ < if(0 ≤ 𝑦, 𝑦, 0))
24		simpll 767	. . . . . . . . . . . . . . . . 17 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → +∞ ∈ 𝑢)
25		simprr 773	. . . . . . . . . . . . . . . . 17 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → 𝑢 = (𝑦(,]+∞))
26	24, 25	eleqtrd 2833	. . . . . . . . . . . . . . . 16 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → +∞ ∈ (𝑦(,]+∞))
27		elioc1 12942	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℝ^* ∧ +∞ ∈ ℝ^) → (+∞ ∈ (𝑦(,]+∞) ↔ (+∞ ∈ ℝ^ ∧ 𝑦 < +∞ ∧ +∞ ≤ +∞)))
28	14, 18, 27	sylancl 589	. . . . . . . . . . . . . . . 16 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → (+∞ ∈ (𝑦(,]+∞) ↔ (+∞ ∈ ℝ^* ∧ 𝑦 < +∞ ∧ +∞ ≤ +∞)))
29	26, 28	mpbid 235	. . . . . . . . . . . . . . 15 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → (+∞ ∈ ℝ^* ∧ 𝑦 < +∞ ∧ +∞ ≤ +∞))
30	29	simp2d 1145	. . . . . . . . . . . . . 14 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → 𝑦 < +∞)
31		0ltpnf 12679	. . . . . . . . . . . . . 14 ⊢ 0 < +∞
32		breq1 5042	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = if(0 ≤ 𝑦, 𝑦, 0) → (𝑦 < +∞ ↔ if(0 ≤ 𝑦, 𝑦, 0) < +∞))
33		breq1 5042	. . . . . . . . . . . . . . 15 ⊢ (0 = if(0 ≤ 𝑦, 𝑦, 0) → (0 < +∞ ↔ if(0 ≤ 𝑦, 𝑦, 0) < +∞))
34	32, 33	ifboth 4464	. . . . . . . . . . . . . 14 ⊢ ((𝑦 < +∞ ∧ 0 < +∞) → if(0 ≤ 𝑦, 𝑦, 0) < +∞)
35	30, 31, 34	sylancl 589	. . . . . . . . . . . . 13 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → if(0 ≤ 𝑦, 𝑦, 0) < +∞)
36		xrre2 12725	. . . . . . . . . . . . 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 1380	. . . . . . . . . . . 12 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ)
38		xrmax2 12731	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → 𝑦 ≤ if(0 ≤ 𝑦, 𝑦, 0))
39	15, 14, 38	sylancr 590	. . . . . . . . . . . . . 14 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → 𝑦 ≤ if(0 ≤ 𝑦, 𝑦, 0))
40		df-ioc 12905	. . . . . . . . . . . . . . 15 ⊢ (,] = (𝑎 ∈ ℝ^, 𝑏 ∈ ℝ^ ↦ {𝑐 ∈ ℝ^* ∣ (𝑎 < 𝑐 ∧ 𝑐 ≤ 𝑏)})
41		xrlelttr 12711	. . . . . . . . . . . . . . 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 12918	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℝ^* ∧ 𝑦 ≤ if(0 ≤ 𝑦, 𝑦, 0)) → (if(0 ≤ 𝑦, 𝑦, 0)(,]+∞) ⊆ (𝑦(,]+∞))
43	14, 39, 42	syl2anc 587	. . . . . . . . . . . . 13 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → (if(0 ≤ 𝑦, 𝑦, 0)(,]+∞) ⊆ (𝑦(,]+∞))
44		simplr 769	. . . . . . . . . . . . . 14 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → 𝑢 ⊆ 𝐴)
45	25, 44	eqsstrrd 3926	. . . . . . . . . . . . 13 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → (𝑦(,]+∞) ⊆ 𝐴)
46	43, 45	sstrd 3897	. . . . . . . . . . . 12 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → (if(0 ≤ 𝑦, 𝑦, 0)(,]+∞) ⊆ 𝐴)
47		oveq1 7198	. . . . . . . . . . . . . 14 ⊢ (𝑥 = if(0 ≤ 𝑦, 𝑦, 0) → (𝑥(,]+∞) = (if(0 ≤ 𝑦, 𝑦, 0)(,]+∞))
48	47	sseq1d 3918	. . . . . . . . . . . . 13 ⊢ (𝑥 = if(0 ≤ 𝑦, 𝑦, 0) → ((𝑥(,]+∞) ⊆ 𝐴 ↔ (if(0 ≤ 𝑦, 𝑦, 0)(,]+∞) ⊆ 𝐴))
49	48	rspcev 3527	. . . . . . . . . . . 12 ⊢ ((if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ ∧ (if(0 ≤ 𝑦, 𝑦, 0)(,]+∞) ⊆ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴)
50	37, 46, 49	syl2anc 587	. . . . . . . . . . 11 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴)
51	50	rexlimdvaa 3194	. . . . . . . . . 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 220	. . . . . . . 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 5810	. . . . . . . . . 10 ⊢ (𝑢 ∈ V → (𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)) ↔ ∃𝑦 ∈ ℝ^* 𝑢 = (-∞[,)𝑦)))
56	55	elv 3404	. . . . . . . . 9 ⊢ (𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)) ↔ ∃𝑦 ∈ ℝ^* 𝑢 = (-∞[,)𝑦))
57		pnfnlt 12685	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ^* → ¬ +∞ < 𝑦)
58		elico1 12943	. . . . . . . . . . . . . . . 16 ⊢ ((-∞ ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → (+∞ ∈ (-∞[,)𝑦) ↔ (+∞ ∈ ℝ^ ∧ -∞ ≤ +∞ ∧ +∞ < 𝑦)))
59	12, 58	mpan 690	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℝ^* → (+∞ ∈ (-∞[,)𝑦) ↔ (+∞ ∈ ℝ^* ∧ -∞ ≤ +∞ ∧ +∞ < 𝑦)))
60		simp3 1140	. . . . . . . . . . . . . . 15 ⊢ ((+∞ ∈ ℝ^* ∧ -∞ ≤ +∞ ∧ +∞ < 𝑦) → +∞ < 𝑦)
61	59, 60	syl6bi 256	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ^* → (+∞ ∈ (-∞[,)𝑦) → +∞ < 𝑦))
62	57, 61	mtod 201	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ^* → ¬ +∞ ∈ (-∞[,)𝑦))
63		eleq2 2819	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (-∞[,)𝑦) → (+∞ ∈ 𝑢 ↔ +∞ ∈ (-∞[,)𝑦)))
64	63	notbid 321	. . . . . . . . . . . . 13 ⊢ (𝑢 = (-∞[,)𝑦) → (¬ +∞ ∈ 𝑢 ↔ ¬ +∞ ∈ (-∞[,)𝑦)))
65	62, 64	syl5ibrcom 250	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ^* → (𝑢 = (-∞[,)𝑦) → ¬ +∞ ∈ 𝑢))
66	65	rexlimiv 3189	. . . . . . . . . . 11 ⊢ (∃𝑦 ∈ ℝ^* 𝑢 = (-∞[,)𝑦) → ¬ +∞ ∈ 𝑢)
67	66	pm2.21d 121	. . . . . . . . . 10 ⊢ (∃𝑦 ∈ ℝ^* 𝑢 = (-∞[,)𝑦) → (+∞ ∈ 𝑢 → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
68	67	adantrd 495	. . . . . . . . 9 ⊢ (∃𝑦 ∈ ℝ^* 𝑢 = (-∞[,)𝑦) → ((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
69	56, 68	sylbi 220	. . . . . . . 8 ⊢ (𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)) → ((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
70	53, 69	jaoi 857	. . . . . . 7 ⊢ ((𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∨ 𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) → ((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
71	8, 70	sylbi 220	. . . . . 6 ⊢ (𝑢 ∈ (ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) → ((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
72		pnfnre 10839	. . . . . . . . . 10 ⊢ +∞ ∉ ℝ
73	72	neli 3038	. . . . . . . . 9 ⊢ ¬ +∞ ∈ ℝ
74		elssuni 4837	. . . . . . . . . . 11 ⊢ (𝑢 ∈ ran (,) → 𝑢 ⊆ ∪ ran (,))
75		unirnioo 13002	. . . . . . . . . . 11 ⊢ ℝ = ∪ ran (,)
76	74, 75	sseqtrrdi 3938	. . . . . . . . . 10 ⊢ (𝑢 ∈ ran (,) → 𝑢 ⊆ ℝ)
77	76	sseld 3886	. . . . . . . . 9 ⊢ (𝑢 ∈ ran (,) → (+∞ ∈ 𝑢 → +∞ ∈ ℝ))
78	73, 77	mtoi 202	. . . . . . . 8 ⊢ (𝑢 ∈ ran (,) → ¬ +∞ ∈ 𝑢)
79	78	pm2.21d 121	. . . . . . 7 ⊢ (𝑢 ∈ ran (,) → (+∞ ∈ 𝑢 → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
80	79	adantrd 495	. . . . . 6 ⊢ (𝑢 ∈ ran (,) → ((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
81	71, 80	jaoi 857	. . . . 5 ⊢ ((𝑢 ∈ (ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ∨ 𝑢 ∈ ran (,)) → ((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
82	7, 81	sylbi 220	. . . 4 ⊢ (𝑢 ∈ ((ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ∪ ran (,)) → ((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
83	82	rexlimiv 3189	. . 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 584	1 ⊢ ((𝐴 ∈ (ordTop‘ ≤ ) ∧ +∞ ∈ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 847 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 ∃wrex 3052 Vcvv 3398 ∪ cun 3851 ⊆ wss 3853 ifcif 4425 ∪ cuni 4805 class class class wbr 5039 ↦ cmpt 5120 ran crn 5537 ‘cfv 6358 (class class class)co 7191 ℝcr 10693 0cc0 10694 +∞cpnf 10829 -∞cmnf 10830 ℝ^*cxr 10831 < clt 10832 ≤ cle 10833 (,)cioo 12900 (,]cioc 12901 [,)cico 12902 topGenctg 16896 ordTopcordt 16958

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-1o 8180 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-fi 9005 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-ioo 12904 df-ioc 12905 df-ico 12906 df-icc 12907 df-topgen 16902 df-ordt 16960 df-ps 18026 df-tsr 18027 df-top 21745 df-bases 21797

This theorem is referenced by: xrge0tsms 23685 xrlimcnp 25805 xrge0tsmsd 30990 pnfneige0 31569 xlimpnfvlem2 42996

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