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Theorem pnfnei 21830

Description: A neighborhood of +∞ contains an unbounded interval based at a real number. Together with xrtgioo 23416 (which describes neighborhoods of ℝ) and mnfnei 21831, this gives all "negative" topological information ensuring that it is not too fine (and of course iooordt 21827 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 2823	. . . 4 ⊢ ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) = ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞))
2		eqid 2823	. . . 4 ⊢ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)) = ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))
3		eqid 2823	. . . 4 ⊢ ran (,) = ran (,)
4	1, 2, 3	leordtval 21823	. . 3 ⊢ (ordTop‘ ≤ ) = (topGen‘((ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ∪ ran (,)))
5	4	eleq2i 2906	. 2 ⊢ (𝐴 ∈ (ordTop‘ ≤ ) ↔ 𝐴 ∈ (topGen‘((ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ∪ ran (,))))
6		tg2 21575	. . 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 4127	. . . . 5 ⊢ (𝑢 ∈ ((ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ∪ ran (,)) ↔ (𝑢 ∈ (ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ∨ 𝑢 ∈ ran (,)))
8		elun 4127	. . . . . . 7 ⊢ (𝑢 ∈ (ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ↔ (𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∨ 𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))))
9		eqid 2823	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) = (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞))
10	9	elrnmpt 5830	. . . . . . . . . 10 ⊢ (𝑢 ∈ V → (𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ↔ ∃𝑦 ∈ ℝ^* 𝑢 = (𝑦(,]+∞)))
11	10	elv 3501	. . . . . . . . 9 ⊢ (𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ↔ ∃𝑦 ∈ ℝ^* 𝑢 = (𝑦(,]+∞))
12		mnfxr 10700	. . . . . . . . . . . . . 14 ⊢ -∞ ∈ ℝ^*
13	12	a1i 11	. . . . . . . . . . . . 13 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → -∞ ∈ ℝ^*)
14		simprl 769	. . . . . . . . . . . . . 14 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → 𝑦 ∈ ℝ^*)
15		0xr 10690	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ^*
16		ifcl 4513	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℝ^* ∧ 0 ∈ ℝ^) → if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ^)
17	14, 15, 16	sylancl 588	. . . . . . . . . . . . 13 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ^*)
18		pnfxr 10697	. . . . . . . . . . . . . 14 ⊢ +∞ ∈ ℝ^*
19	18	a1i 11	. . . . . . . . . . . . 13 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → +∞ ∈ ℝ^*)
20		xrmax1 12571	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → 0 ≤ if(0 ≤ 𝑦, 𝑦, 0))
21	15, 14, 20	sylancr 589	. . . . . . . . . . . . . 14 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → 0 ≤ if(0 ≤ 𝑦, 𝑦, 0))
22		ge0gtmnf 12568	. . . . . . . . . . . . . 14 ⊢ ((if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ^* ∧ 0 ≤ if(0 ≤ 𝑦, 𝑦, 0)) → -∞ < if(0 ≤ 𝑦, 𝑦, 0))
23	17, 21, 22	syl2anc 586	. . . . . . . . . . . . 13 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → -∞ < if(0 ≤ 𝑦, 𝑦, 0))
24		simpll 765	. . . . . . . . . . . . . . . . 17 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → +∞ ∈ 𝑢)
25		simprr 771	. . . . . . . . . . . . . . . . 17 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → 𝑢 = (𝑦(,]+∞))
26	24, 25	eleqtrd 2917	. . . . . . . . . . . . . . . 16 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → +∞ ∈ (𝑦(,]+∞))
27		elioc1 12783	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℝ^* ∧ +∞ ∈ ℝ^) → (+∞ ∈ (𝑦(,]+∞) ↔ (+∞ ∈ ℝ^ ∧ 𝑦 < +∞ ∧ +∞ ≤ +∞)))
28	14, 18, 27	sylancl 588	. . . . . . . . . . . . . . . 16 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → (+∞ ∈ (𝑦(,]+∞) ↔ (+∞ ∈ ℝ^* ∧ 𝑦 < +∞ ∧ +∞ ≤ +∞)))
29	26, 28	mpbid 234	. . . . . . . . . . . . . . 15 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → (+∞ ∈ ℝ^* ∧ 𝑦 < +∞ ∧ +∞ ≤ +∞))
30	29	simp2d 1139	. . . . . . . . . . . . . 14 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → 𝑦 < +∞)
31		0ltpnf 12520	. . . . . . . . . . . . . 14 ⊢ 0 < +∞
32		breq1 5071	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = if(0 ≤ 𝑦, 𝑦, 0) → (𝑦 < +∞ ↔ if(0 ≤ 𝑦, 𝑦, 0) < +∞))
33		breq1 5071	. . . . . . . . . . . . . . 15 ⊢ (0 = if(0 ≤ 𝑦, 𝑦, 0) → (0 < +∞ ↔ if(0 ≤ 𝑦, 𝑦, 0) < +∞))
34	32, 33	ifboth 4507	. . . . . . . . . . . . . 14 ⊢ ((𝑦 < +∞ ∧ 0 < +∞) → if(0 ≤ 𝑦, 𝑦, 0) < +∞)
35	30, 31, 34	sylancl 588	. . . . . . . . . . . . 13 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → if(0 ≤ 𝑦, 𝑦, 0) < +∞)
36		xrre2 12566	. . . . . . . . . . . . 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 1374	. . . . . . . . . . . 12 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ)
38		xrmax2 12572	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → 𝑦 ≤ if(0 ≤ 𝑦, 𝑦, 0))
39	15, 14, 38	sylancr 589	. . . . . . . . . . . . . 14 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → 𝑦 ≤ if(0 ≤ 𝑦, 𝑦, 0))
40		df-ioc 12746	. . . . . . . . . . . . . . 15 ⊢ (,] = (𝑎 ∈ ℝ^, 𝑏 ∈ ℝ^ ↦ {𝑐 ∈ ℝ^* ∣ (𝑎 < 𝑐 ∧ 𝑐 ≤ 𝑏)})
41		xrlelttr 12552	. . . . . . . . . . . . . . 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 12759	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℝ^* ∧ 𝑦 ≤ if(0 ≤ 𝑦, 𝑦, 0)) → (if(0 ≤ 𝑦, 𝑦, 0)(,]+∞) ⊆ (𝑦(,]+∞))
43	14, 39, 42	syl2anc 586	. . . . . . . . . . . . 13 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → (if(0 ≤ 𝑦, 𝑦, 0)(,]+∞) ⊆ (𝑦(,]+∞))
44		simplr 767	. . . . . . . . . . . . . 14 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → 𝑢 ⊆ 𝐴)
45	25, 44	eqsstrrd 4008	. . . . . . . . . . . . 13 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → (𝑦(,]+∞) ⊆ 𝐴)
46	43, 45	sstrd 3979	. . . . . . . . . . . 12 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → (if(0 ≤ 𝑦, 𝑦, 0)(,]+∞) ⊆ 𝐴)
47		oveq1 7165	. . . . . . . . . . . . . 14 ⊢ (𝑥 = if(0 ≤ 𝑦, 𝑦, 0) → (𝑥(,]+∞) = (if(0 ≤ 𝑦, 𝑦, 0)(,]+∞))
48	47	sseq1d 4000	. . . . . . . . . . . . 13 ⊢ (𝑥 = if(0 ≤ 𝑦, 𝑦, 0) → ((𝑥(,]+∞) ⊆ 𝐴 ↔ (if(0 ≤ 𝑦, 𝑦, 0)(,]+∞) ⊆ 𝐴))
49	48	rspcev 3625	. . . . . . . . . . . 12 ⊢ ((if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ ∧ (if(0 ≤ 𝑦, 𝑦, 0)(,]+∞) ⊆ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴)
50	37, 46, 49	syl2anc 586	. . . . . . . . . . 11 ⊢ (((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (𝑦(,]+∞))) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴)
51	50	rexlimdvaa 3287	. . . . . . . . . 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 219	. . . . . . . 8 ⊢ (𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) → ((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
54		eqid 2823	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)) = (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))
55	54	elrnmpt 5830	. . . . . . . . . 10 ⊢ (𝑢 ∈ V → (𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)) ↔ ∃𝑦 ∈ ℝ^* 𝑢 = (-∞[,)𝑦)))
56	55	elv 3501	. . . . . . . . 9 ⊢ (𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)) ↔ ∃𝑦 ∈ ℝ^* 𝑢 = (-∞[,)𝑦))
57		pnfnlt 12526	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ^* → ¬ +∞ < 𝑦)
58		elico1 12784	. . . . . . . . . . . . . . . 16 ⊢ ((-∞ ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → (+∞ ∈ (-∞[,)𝑦) ↔ (+∞ ∈ ℝ^ ∧ -∞ ≤ +∞ ∧ +∞ < 𝑦)))
59	12, 58	mpan 688	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℝ^* → (+∞ ∈ (-∞[,)𝑦) ↔ (+∞ ∈ ℝ^* ∧ -∞ ≤ +∞ ∧ +∞ < 𝑦)))
60		simp3 1134	. . . . . . . . . . . . . . 15 ⊢ ((+∞ ∈ ℝ^* ∧ -∞ ≤ +∞ ∧ +∞ < 𝑦) → +∞ < 𝑦)
61	59, 60	syl6bi 255	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ^* → (+∞ ∈ (-∞[,)𝑦) → +∞ < 𝑦))
62	57, 61	mtod 200	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ^* → ¬ +∞ ∈ (-∞[,)𝑦))
63		eleq2 2903	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (-∞[,)𝑦) → (+∞ ∈ 𝑢 ↔ +∞ ∈ (-∞[,)𝑦)))
64	63	notbid 320	. . . . . . . . . . . . 13 ⊢ (𝑢 = (-∞[,)𝑦) → (¬ +∞ ∈ 𝑢 ↔ ¬ +∞ ∈ (-∞[,)𝑦)))
65	62, 64	syl5ibrcom 249	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ^* → (𝑢 = (-∞[,)𝑦) → ¬ +∞ ∈ 𝑢))
66	65	rexlimiv 3282	. . . . . . . . . . 11 ⊢ (∃𝑦 ∈ ℝ^* 𝑢 = (-∞[,)𝑦) → ¬ +∞ ∈ 𝑢)
67	66	pm2.21d 121	. . . . . . . . . 10 ⊢ (∃𝑦 ∈ ℝ^* 𝑢 = (-∞[,)𝑦) → (+∞ ∈ 𝑢 → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
68	67	adantrd 494	. . . . . . . . 9 ⊢ (∃𝑦 ∈ ℝ^* 𝑢 = (-∞[,)𝑦) → ((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
69	56, 68	sylbi 219	. . . . . . . 8 ⊢ (𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)) → ((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
70	53, 69	jaoi 853	. . . . . . 7 ⊢ ((𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∨ 𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) → ((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
71	8, 70	sylbi 219	. . . . . 6 ⊢ (𝑢 ∈ (ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) → ((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
72		pnfnre 10684	. . . . . . . . . 10 ⊢ +∞ ∉ ℝ
73	72	neli 3127	. . . . . . . . 9 ⊢ ¬ +∞ ∈ ℝ
74		elssuni 4870	. . . . . . . . . . 11 ⊢ (𝑢 ∈ ran (,) → 𝑢 ⊆ ∪ ran (,))
75		unirnioo 12840	. . . . . . . . . . 11 ⊢ ℝ = ∪ ran (,)
76	74, 75	sseqtrrdi 4020	. . . . . . . . . 10 ⊢ (𝑢 ∈ ran (,) → 𝑢 ⊆ ℝ)
77	76	sseld 3968	. . . . . . . . 9 ⊢ (𝑢 ∈ ran (,) → (+∞ ∈ 𝑢 → +∞ ∈ ℝ))
78	73, 77	mtoi 201	. . . . . . . 8 ⊢ (𝑢 ∈ ran (,) → ¬ +∞ ∈ 𝑢)
79	78	pm2.21d 121	. . . . . . 7 ⊢ (𝑢 ∈ ran (,) → (+∞ ∈ 𝑢 → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
80	79	adantrd 494	. . . . . 6 ⊢ (𝑢 ∈ ran (,) → ((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
81	71, 80	jaoi 853	. . . . 5 ⊢ ((𝑢 ∈ (ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ∨ 𝑢 ∈ ran (,)) → ((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
82	7, 81	sylbi 219	. . . 4 ⊢ (𝑢 ∈ ((ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ∪ ran (,)) → ((+∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
83	82	rexlimiv 3282	. . 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 583	1 ⊢ ((𝐴 ∈ (ordTop‘ ≤ ) ∧ +∞ ∈ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∃wrex 3141 Vcvv 3496 ∪ cun 3936 ⊆ wss 3938 ifcif 4469 ∪ cuni 4840 class class class wbr 5068 ↦ cmpt 5148 ran crn 5558 ‘cfv 6357 (class class class)co 7158 ℝcr 10538 0cc0 10539 +∞cpnf 10674 -∞cmnf 10675 ℝ^*cxr 10676 < clt 10677 ≤ cle 10678 (,)cioo 12741 (,]cioc 12742 [,)cico 12743 topGenctg 16713 ordTopcordt 16774

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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fi 8877 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-ioo 12745 df-ioc 12746 df-ico 12747 df-icc 12748 df-topgen 16719 df-ordt 16776 df-ps 17812 df-tsr 17813 df-top 21504 df-bases 21556

This theorem is referenced by: xrge0tsms 23444 xrlimcnp 25548 xrge0tsmsd 30694 pnfneige0 31196 xlimpnfvlem2 42125

W3C validator