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Theorem pnfnei 23083

Description: A neighborhood of +∞ contains an unbounded interval based at a real number. Together with xrtgioo 24671 (which describes neighborhoods of ℝ) and mnfnei 23084, this gives all "negative" topological information ensuring that it is not too fine (and of course iooordt 23080 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 2729	. . . 4 ⊢ ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) = ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞))
2		eqid 2729	. . . 4 ⊢ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)) = ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))
3		eqid 2729	. . . 4 ⊢ ran (,) = ran (,)
4	1, 2, 3	leordtval 23076	. . 3 ⊢ (ordTop‘ ≤ ) = (topGen‘((ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ∪ ran (,)))
5	4	eleq2i 2820	. 2 ⊢ (𝐴 ∈ (ordTop‘ ≤ ) ↔ 𝐴 ∈ (topGen‘((ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ∪ ran (,))))
6		tg2 22828	. . 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 4112	. . . . 5 ⊢ (𝑢 ∈ ((ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ∪ ran (,)) ↔ (𝑢 ∈ (ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ∨ 𝑢 ∈ ran (,)))
8		elun 4112	. . . . . . 7 ⊢ (𝑢 ∈ (ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ↔ (𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∨ 𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))))
9		eqid 2729	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) = (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞))
10	9	elrnmpt 5911	. . . . . . . . . 10 ⊢ (𝑢 ∈ V → (𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ↔ ∃𝑦 ∈ ℝ^* 𝑢 = (𝑦(,]+∞)))
11	10	elv 3449	. . . . . . . . 9 ⊢ (𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ↔ ∃𝑦 ∈ ℝ^* 𝑢 = (𝑦(,]+∞))
12		mnfxr 11207	. . . . . . . . . . . . . 14 ⊢ -∞ ∈ ℝ^*
13	12	a1i 11	. . . . . . . . . . . . 13 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → -∞ ∈ ℝ^*)
14		simprl 770	. . . . . . . . . . . . . 14 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → 𝑦 ∈ ℝ^*)
15		0xr 11197	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ^*
16		ifcl 4530	. . . . . . . . . . . . . 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 11204	. . . . . . . . . . . . . 14 ⊢ +∞ ∈ ℝ^*
19	18	a1i 11	. . . . . . . . . . . . 13 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → +∞ ∈ ℝ^*)
20		xrmax1 13111	. . . . . . . . . . . . . . 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 13108	. . . . . . . . . . . . . 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 766	. . . . . . . . . . . . . . . . 17 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → +∞ ∈ 𝑢)
25		simprr 772	. . . . . . . . . . . . . . . . 17 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → 𝑢 = (𝑦(,]+∞))
26	24, 25	eleqtrd 2830	. . . . . . . . . . . . . . . 16 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → +∞ ∈ (𝑦(,]+∞))
27		elioc1 13324	. . . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . . 15 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → (+∞ ∈ ℝ^* ∧ 𝑦 < +∞ ∧ +∞ ≤ +∞))
30	29	simp2d 1143	. . . . . . . . . . . . . 14 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → 𝑦 < +∞)
31		0ltpnf 13058	. . . . . . . . . . . . . 14 ⊢ 0 < +∞
32		breq1 5105	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = if(0 ≤ 𝑦, 𝑦, 0) → (𝑦 < +∞ ↔ if(0 ≤ 𝑦, 𝑦, 0) < +∞))
33		breq1 5105	. . . . . . . . . . . . . . 15 ⊢ (0 = if(0 ≤ 𝑦, 𝑦, 0) → (0 < +∞ ↔ if(0 ≤ 𝑦, 𝑦, 0) < +∞))
34	32, 33	ifboth 4524	. . . . . . . . . . . . . 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 13106	. . . . . . . . . . . . 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 13112	. . . . . . . . . . . . . . 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 13287	. . . . . . . . . . . . . . 15 ⊢ (,] = (𝑎 ∈ ℝ^, 𝑏 ∈ ℝ^ ↦ {𝑐 ∈ ℝ^* ∣ (𝑎 < 𝑐 ∧ 𝑐 ≤ 𝑏)})
41		xrlelttr 13092	. . . . . . . . . . . . . . 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 13300	. . . . . . . . . . . . . 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 768	. . . . . . . . . . . . . 14 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → 𝑢 ⊆ 𝐴)
45	25, 44	eqsstrrd 3979	. . . . . . . . . . . . 13 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → (𝑦(,]+∞) ⊆ 𝐴)
46	43, 45	sstrd 3954	. . . . . . . . . . . 12 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → (if(0 ≤ 𝑦, 𝑦, 0)(,]+∞) ⊆ 𝐴)
47		oveq1 7376	. . . . . . . . . . . . . 14 ⊢ (𝑥 = if(0 ≤ 𝑦, 𝑦, 0) → (𝑥(,]+∞) = (if(0 ≤ 𝑦, 𝑦, 0)(,]+∞))
48	47	sseq1d 3975	. . . . . . . . . . . . 13 ⊢ (𝑥 = if(0 ≤ 𝑦, 𝑦, 0) → ((𝑥(,]+∞) ⊆ 𝐴 ↔ (if(0 ≤ 𝑦, 𝑦, 0)(,]+∞) ⊆ 𝐴))
49	48	rspcev 3585	. . . . . . . . . . . 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 3135	. . . . . . . . . 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 217	. . . . . . . 8 ⊢ (𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) → ((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
54		eqid 2729	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)) = (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))
55	54	elrnmpt 5911	. . . . . . . . . 10 ⊢ (𝑢 ∈ V → (𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)) ↔ ∃𝑦 ∈ ℝ^* 𝑢 = (-∞[,)𝑦)))
56	55	elv 3449	. . . . . . . . 9 ⊢ (𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)) ↔ ∃𝑦 ∈ ℝ^* 𝑢 = (-∞[,)𝑦))
57		pnfnlt 13064	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ^* → ¬ +∞ < 𝑦)
58		elico1 13325	. . . . . . . . . . . . . . . 16 ⊢ ((-∞ ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → (+∞ ∈ (-∞[,)𝑦) ↔ (+∞ ∈ ℝ^ ∧ -∞ ≤ +∞ ∧ +∞ < 𝑦)))
59	12, 58	mpan 690	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℝ^* → (+∞ ∈ (-∞[,)𝑦) ↔ (+∞ ∈ ℝ^* ∧ -∞ ≤ +∞ ∧ +∞ < 𝑦)))
60		simp3 1138	. . . . . . . . . . . . . . 15 ⊢ ((+∞ ∈ ℝ^* ∧ -∞ ≤ +∞ ∧ +∞ < 𝑦) → +∞ < 𝑦)
61	59, 60	biimtrdi 253	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ^* → (+∞ ∈ (-∞[,)𝑦) → +∞ < 𝑦))
62	57, 61	mtod 198	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ^* → ¬ +∞ ∈ (-∞[,)𝑦))
63		eleq2 2817	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (-∞[,)𝑦) → (+∞ ∈ 𝑢 ↔ +∞ ∈ (-∞[,)𝑦)))
64	63	notbid 318	. . . . . . . . . . . . 13 ⊢ (𝑢 = (-∞[,)𝑦) → (¬ +∞ ∈ 𝑢 ↔ ¬ +∞ ∈ (-∞[,)𝑦)))
65	62, 64	syl5ibrcom 247	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ^* → (𝑢 = (-∞[,)𝑦) → ¬ +∞ ∈ 𝑢))
66	65	rexlimiv 3127	. . . . . . . . . . 11 ⊢ (∃𝑦 ∈ ℝ^* 𝑢 = (-∞[,)𝑦) → ¬ +∞ ∈ 𝑢)
67	66	pm2.21d 121	. . . . . . . . . 10 ⊢ (∃𝑦 ∈ ℝ^* 𝑢 = (-∞[,)𝑦) → (+∞ ∈ 𝑢 → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
68	67	adantrd 491	. . . . . . . . 9 ⊢ (∃𝑦 ∈ ℝ^* 𝑢 = (-∞[,)𝑦) → ((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
69	56, 68	sylbi 217	. . . . . . . 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 217	. . . . . 6 ⊢ (𝑢 ∈ (ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) → ((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
72		pnfnre 11191	. . . . . . . . . 10 ⊢ +∞ ∉ ℝ
73	72	neli 3031	. . . . . . . . 9 ⊢ ¬ +∞ ∈ ℝ
74		elssuni 4897	. . . . . . . . . . 11 ⊢ (𝑢 ∈ ran (,) → 𝑢 ⊆ ∪ ran (,))
75		unirnioo 13386	. . . . . . . . . . 11 ⊢ ℝ = ∪ ran (,)
76	74, 75	sseqtrrdi 3985	. . . . . . . . . 10 ⊢ (𝑢 ∈ ran (,) → 𝑢 ⊆ ℝ)
77	76	sseld 3942	. . . . . . . . 9 ⊢ (𝑢 ∈ ran (,) → (+∞ ∈ 𝑢 → +∞ ∈ ℝ))
78	73, 77	mtoi 199	. . . . . . . 8 ⊢ (𝑢 ∈ ran (,) → ¬ +∞ ∈ 𝑢)
79	78	pm2.21d 121	. . . . . . 7 ⊢ (𝑢 ∈ ran (,) → (+∞ ∈ 𝑢 → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
80	79	adantrd 491	. . . . . 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 217	. . . 4 ⊢ (𝑢 ∈ ((ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ∪ ran (,)) → ((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
83	82	rexlimiv 3127	. . 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 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∃wrex 3053 Vcvv 3444 ∪ cun 3909 ⊆ wss 3911 ifcif 4484 ∪ cuni 4867 class class class wbr 5102 ↦ cmpt 5183 ran crn 5632 ‘cfv 6499 (class class class)co 7369 ℝcr 11043 0cc0 11044 +∞cpnf 11181 -∞cmnf 11182 ℝ^*cxr 11183 < clt 11184 ≤ cle 11185 (,)cioo 13282 (,]cioc 13283 [,)cico 13284 topGenctg 17376 ordTopcordt 17438

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-om 7823 df-1st 7947 df-2nd 7948 df-1o 8411 df-2o 8412 df-er 8648 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fi 9338 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-ioo 13286 df-ioc 13287 df-ico 13288 df-icc 13289 df-topgen 17382 df-ordt 17440 df-ps 18501 df-tsr 18502 df-top 22757 df-bases 22809

This theorem is referenced by: xrge0tsms 24699 xrlimcnp 26854 xrge0tsmsd 32975 pnfneige0 33914 xlimpnfvlem2 45808

W3C validator