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Theorem mnfnei 22372

Description: A neighborhood of -∞ contains an unbounded interval based at a real number. (Contributed by Mario Carneiro, 3-Sep-2015.)

**Assertion**
Ref	Expression
mnfnei	⊢ ((𝐴 ∈ (ordTop‘ ≤ ) ∧ -∞ ∈ 𝐴) → ∃𝑥 ∈ ℝ (-∞[,)𝑥) ⊆ 𝐴)

Distinct variable group: 𝑥,𝐴

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

Step	Hyp	Ref	Expression
1		eqid 2738	. . . 4 ⊢ ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) = ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞))
2		eqid 2738	. . . 4 ⊢ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)) = ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))
3		eqid 2738	. . . 4 ⊢ ran (,) = ran (,)
4	1, 2, 3	leordtval 22364	. . 3 ⊢ (ordTop‘ ≤ ) = (topGen‘((ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ∪ ran (,)))
5	4	eleq2i 2830	. 2 ⊢ (𝐴 ∈ (ordTop‘ ≤ ) ↔ 𝐴 ∈ (topGen‘((ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ∪ ran (,))))
6		tg2 22115	. . 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 4083	. . . . 5 ⊢ (𝑢 ∈ ((ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ∪ ran (,)) ↔ (𝑢 ∈ (ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ∨ 𝑢 ∈ ran (,)))
8		elun 4083	. . . . . . 7 ⊢ (𝑢 ∈ (ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ↔ (𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∨ 𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))))
9		eqid 2738	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) = (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞))
10	9	elrnmpt 5865	. . . . . . . . . 10 ⊢ (𝑢 ∈ V → (𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ↔ ∃𝑦 ∈ ℝ^* 𝑢 = (𝑦(,]+∞)))
11	10	elv 3438	. . . . . . . . 9 ⊢ (𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ↔ ∃𝑦 ∈ ℝ^* 𝑢 = (𝑦(,]+∞))
12		nltmnf 12865	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ^* → ¬ 𝑦 < -∞)
13		pnfxr 11029	. . . . . . . . . . . . . . . 16 ⊢ +∞ ∈ ℝ^*
14		elioc1 13121	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℝ^* ∧ +∞ ∈ ℝ^) → (-∞ ∈ (𝑦(,]+∞) ↔ (-∞ ∈ ℝ^ ∧ 𝑦 < -∞ ∧ -∞ ≤ +∞)))
15	13, 14	mpan2 688	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℝ^* → (-∞ ∈ (𝑦(,]+∞) ↔ (-∞ ∈ ℝ^* ∧ 𝑦 < -∞ ∧ -∞ ≤ +∞)))
16		simp2 1136	. . . . . . . . . . . . . . 15 ⊢ ((-∞ ∈ ℝ^* ∧ 𝑦 < -∞ ∧ -∞ ≤ +∞) → 𝑦 < -∞)
17	15, 16	syl6bi 252	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ^* → (-∞ ∈ (𝑦(,]+∞) → 𝑦 < -∞))
18	12, 17	mtod 197	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ^* → ¬ -∞ ∈ (𝑦(,]+∞))
19		eleq2 2827	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (𝑦(,]+∞) → (-∞ ∈ 𝑢 ↔ -∞ ∈ (𝑦(,]+∞)))
20	19	notbid 318	. . . . . . . . . . . . 13 ⊢ (𝑢 = (𝑦(,]+∞) → (¬ -∞ ∈ 𝑢 ↔ ¬ -∞ ∈ (𝑦(,]+∞)))
21	18, 20	syl5ibrcom 246	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ^* → (𝑢 = (𝑦(,]+∞) → ¬ -∞ ∈ 𝑢))
22	21	rexlimiv 3209	. . . . . . . . . . 11 ⊢ (∃𝑦 ∈ ℝ^* 𝑢 = (𝑦(,]+∞) → ¬ -∞ ∈ 𝑢)
23	22	pm2.21d 121	. . . . . . . . . 10 ⊢ (∃𝑦 ∈ ℝ^* 𝑢 = (𝑦(,]+∞) → (-∞ ∈ 𝑢 → ∃𝑥 ∈ ℝ (-∞[,)𝑥) ⊆ 𝐴))
24	23	adantrd 492	. . . . . . . . 9 ⊢ (∃𝑦 ∈ ℝ^* 𝑢 = (𝑦(,]+∞) → ((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (-∞[,)𝑥) ⊆ 𝐴))
25	11, 24	sylbi 216	. . . . . . . 8 ⊢ (𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) → ((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (-∞[,)𝑥) ⊆ 𝐴))
26		eqid 2738	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)) = (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))
27	26	elrnmpt 5865	. . . . . . . . . 10 ⊢ (𝑢 ∈ V → (𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)) ↔ ∃𝑦 ∈ ℝ^* 𝑢 = (-∞[,)𝑦)))
28	27	elv 3438	. . . . . . . . 9 ⊢ (𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)) ↔ ∃𝑦 ∈ ℝ^* 𝑢 = (-∞[,)𝑦))
29		mnfxr 11032	. . . . . . . . . . . . . 14 ⊢ -∞ ∈ ℝ^*
30	29	a1i 11	. . . . . . . . . . . . 13 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → -∞ ∈ ℝ^*)
31		0xr 11022	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ^*
32		simprl 768	. . . . . . . . . . . . . 14 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → 𝑦 ∈ ℝ^*)
33		ifcl 4504	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → if(0 ≤ 𝑦, 0, 𝑦) ∈ ℝ^)
34	31, 32, 33	sylancr 587	. . . . . . . . . . . . 13 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → if(0 ≤ 𝑦, 0, 𝑦) ∈ ℝ^*)
35	13	a1i 11	. . . . . . . . . . . . 13 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → +∞ ∈ ℝ^*)
36		mnflt0 12861	. . . . . . . . . . . . . 14 ⊢ -∞ < 0
37		simpll 764	. . . . . . . . . . . . . . . . 17 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → -∞ ∈ 𝑢)
38		simprr 770	. . . . . . . . . . . . . . . . 17 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → 𝑢 = (-∞[,)𝑦))
39	37, 38	eleqtrd 2841	. . . . . . . . . . . . . . . 16 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → -∞ ∈ (-∞[,)𝑦))
40		elico1 13122	. . . . . . . . . . . . . . . . 17 ⊢ ((-∞ ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → (-∞ ∈ (-∞[,)𝑦) ↔ (-∞ ∈ ℝ^ ∧ -∞ ≤ -∞ ∧ -∞ < 𝑦)))
41	29, 32, 40	sylancr 587	. . . . . . . . . . . . . . . 16 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → (-∞ ∈ (-∞[,)𝑦) ↔ (-∞ ∈ ℝ^* ∧ -∞ ≤ -∞ ∧ -∞ < 𝑦)))
42	39, 41	mpbid 231	. . . . . . . . . . . . . . 15 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → (-∞ ∈ ℝ^* ∧ -∞ ≤ -∞ ∧ -∞ < 𝑦))
43	42	simp3d 1143	. . . . . . . . . . . . . 14 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → -∞ < 𝑦)
44		breq2 5078	. . . . . . . . . . . . . . 15 ⊢ (0 = if(0 ≤ 𝑦, 0, 𝑦) → (-∞ < 0 ↔ -∞ < if(0 ≤ 𝑦, 0, 𝑦)))
45		breq2 5078	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = if(0 ≤ 𝑦, 0, 𝑦) → (-∞ < 𝑦 ↔ -∞ < if(0 ≤ 𝑦, 0, 𝑦)))
46	44, 45	ifboth 4498	. . . . . . . . . . . . . 14 ⊢ ((-∞ < 0 ∧ -∞ < 𝑦) → -∞ < if(0 ≤ 𝑦, 0, 𝑦))
47	36, 43, 46	sylancr 587	. . . . . . . . . . . . 13 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → -∞ < if(0 ≤ 𝑦, 0, 𝑦))
48	31	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → 0 ∈ ℝ^*)
49		xrmin1 12911	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → if(0 ≤ 𝑦, 0, 𝑦) ≤ 0)
50	31, 32, 49	sylancr 587	. . . . . . . . . . . . . 14 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → if(0 ≤ 𝑦, 0, 𝑦) ≤ 0)
51		0re 10977	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℝ
52		ltpnf 12856	. . . . . . . . . . . . . . 15 ⊢ (0 ∈ ℝ → 0 < +∞)
53	51, 52	mp1i 13	. . . . . . . . . . . . . 14 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → 0 < +∞)
54	34, 48, 35, 50, 53	xrlelttrd 12894	. . . . . . . . . . . . 13 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → if(0 ≤ 𝑦, 0, 𝑦) < +∞)
55		xrre2 12904	. . . . . . . . . . . . 13 ⊢ (((-∞ ∈ ℝ^* ∧ if(0 ≤ 𝑦, 0, 𝑦) ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (-∞ < if(0 ≤ 𝑦, 0, 𝑦) ∧ if(0 ≤ 𝑦, 0, 𝑦) < +∞)) → if(0 ≤ 𝑦, 0, 𝑦) ∈ ℝ)
56	30, 34, 35, 47, 54, 55	syl32anc 1377	. . . . . . . . . . . 12 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → if(0 ≤ 𝑦, 0, 𝑦) ∈ ℝ)
57		xrmin2 12912	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → if(0 ≤ 𝑦, 0, 𝑦) ≤ 𝑦)
58	31, 32, 57	sylancr 587	. . . . . . . . . . . . . 14 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → if(0 ≤ 𝑦, 0, 𝑦) ≤ 𝑦)
59		df-ico 13085	. . . . . . . . . . . . . . 15 ⊢ [,) = (𝑎 ∈ ℝ^, 𝑏 ∈ ℝ^ ↦ {𝑐 ∈ ℝ^* ∣ (𝑎 ≤ 𝑐 ∧ 𝑐 < 𝑏)})
60		xrltletr 12891	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ^* ∧ if(0 ≤ 𝑦, 0, 𝑦) ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → ((𝑥 < if(0 ≤ 𝑦, 0, 𝑦) ∧ if(0 ≤ 𝑦, 0, 𝑦) ≤ 𝑦) → 𝑥 < 𝑦))
61	59, 59, 60	ixxss2 13098	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℝ^* ∧ if(0 ≤ 𝑦, 0, 𝑦) ≤ 𝑦) → (-∞[,)if(0 ≤ 𝑦, 0, 𝑦)) ⊆ (-∞[,)𝑦))
62	32, 58, 61	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → (-∞[,)if(0 ≤ 𝑦, 0, 𝑦)) ⊆ (-∞[,)𝑦))
63		simplr 766	. . . . . . . . . . . . . 14 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → 𝑢 ⊆ 𝐴)
64	38, 63	eqsstrrd 3960	. . . . . . . . . . . . 13 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → (-∞[,)𝑦) ⊆ 𝐴)
65	62, 64	sstrd 3931	. . . . . . . . . . . 12 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → (-∞[,)if(0 ≤ 𝑦, 0, 𝑦)) ⊆ 𝐴)
66		oveq2 7283	. . . . . . . . . . . . . 14 ⊢ (𝑥 = if(0 ≤ 𝑦, 0, 𝑦) → (-∞[,)𝑥) = (-∞[,)if(0 ≤ 𝑦, 0, 𝑦)))
67	66	sseq1d 3952	. . . . . . . . . . . . 13 ⊢ (𝑥 = if(0 ≤ 𝑦, 0, 𝑦) → ((-∞[,)𝑥) ⊆ 𝐴 ↔ (-∞[,)if(0 ≤ 𝑦, 0, 𝑦)) ⊆ 𝐴))
68	67	rspcev 3561	. . . . . . . . . . . 12 ⊢ ((if(0 ≤ 𝑦, 0, 𝑦) ∈ ℝ ∧ (-∞[,)if(0 ≤ 𝑦, 0, 𝑦)) ⊆ 𝐴) → ∃𝑥 ∈ ℝ (-∞[,)𝑥) ⊆ 𝐴)
69	56, 65, 68	syl2anc 584	. . . . . . . . . . 11 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → ∃𝑥 ∈ ℝ (-∞[,)𝑥) ⊆ 𝐴)
70	69	rexlimdvaa 3214	. . . . . . . . . 10 ⊢ ((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → (∃𝑦 ∈ ℝ^* 𝑢 = (-∞[,)𝑦) → ∃𝑥 ∈ ℝ (-∞[,)𝑥) ⊆ 𝐴))
71	70	com12 32	. . . . . . . . 9 ⊢ (∃𝑦 ∈ ℝ^* 𝑢 = (-∞[,)𝑦) → ((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (-∞[,)𝑥) ⊆ 𝐴))
72	28, 71	sylbi 216	. . . . . . . 8 ⊢ (𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)) → ((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (-∞[,)𝑥) ⊆ 𝐴))
73	25, 72	jaoi 854	. . . . . . 7 ⊢ ((𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∨ 𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) → ((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (-∞[,)𝑥) ⊆ 𝐴))
74	8, 73	sylbi 216	. . . . . 6 ⊢ (𝑢 ∈ (ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) → ((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (-∞[,)𝑥) ⊆ 𝐴))
75		mnfnre 11018	. . . . . . . . . 10 ⊢ -∞ ∉ ℝ
76	75	neli 3051	. . . . . . . . 9 ⊢ ¬ -∞ ∈ ℝ
77		elssuni 4871	. . . . . . . . . . 11 ⊢ (𝑢 ∈ ran (,) → 𝑢 ⊆ ∪ ran (,))
78		unirnioo 13181	. . . . . . . . . . 11 ⊢ ℝ = ∪ ran (,)
79	77, 78	sseqtrrdi 3972	. . . . . . . . . 10 ⊢ (𝑢 ∈ ran (,) → 𝑢 ⊆ ℝ)
80	79	sseld 3920	. . . . . . . . 9 ⊢ (𝑢 ∈ ran (,) → (-∞ ∈ 𝑢 → -∞ ∈ ℝ))
81	76, 80	mtoi 198	. . . . . . . 8 ⊢ (𝑢 ∈ ran (,) → ¬ -∞ ∈ 𝑢)
82	81	pm2.21d 121	. . . . . . 7 ⊢ (𝑢 ∈ ran (,) → (-∞ ∈ 𝑢 → ∃𝑥 ∈ ℝ (-∞[,)𝑥) ⊆ 𝐴))
83	82	adantrd 492	. . . . . 6 ⊢ (𝑢 ∈ ran (,) → ((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (-∞[,)𝑥) ⊆ 𝐴))
84	74, 83	jaoi 854	. . . . 5 ⊢ ((𝑢 ∈ (ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ∨ 𝑢 ∈ ran (,)) → ((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (-∞[,)𝑥) ⊆ 𝐴))
85	7, 84	sylbi 216	. . . 4 ⊢ (𝑢 ∈ ((ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ∪ ran (,)) → ((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (-∞[,)𝑥) ⊆ 𝐴))
86	85	rexlimiv 3209	. . 3 ⊢ (∃𝑢 ∈ ((ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ∪ ran (,))(-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (-∞[,)𝑥) ⊆ 𝐴)
87	6, 86	syl 17	. 2 ⊢ ((𝐴 ∈ (topGen‘((ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ∪ ran (,))) ∧ -∞ ∈ 𝐴) → ∃𝑥 ∈ ℝ (-∞[,)𝑥) ⊆ 𝐴)
88	5, 87	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 1539 ∈ wcel 2106 ∃wrex 3065 Vcvv 3432 ∪ cun 3885 ⊆ wss 3887 ifcif 4459 ∪ cuni 4839 class class class wbr 5074 ↦ cmpt 5157 ran crn 5590 ‘cfv 6433 (class class class)co 7275 ℝcr 10870 0cc0 10871 +∞cpnf 11006 -∞cmnf 11007 ℝ^*cxr 11008 < clt 11009 ≤ cle 11010 (,)cioo 13079 (,]cioc 13080 [,)cico 13081 topGenctg 17148 ordTopcordt 17210

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-1o 8297 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fi 9170 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-ioo 13083 df-ioc 13084 df-ico 13085 df-icc 13086 df-topgen 17154 df-ordt 17212 df-ps 18284 df-tsr 18285 df-top 22043 df-bases 22096

This theorem is referenced by: xlimmnfvlem2 43374

W3C validator