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Theorem mnfnei 23186

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 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 23178	. . 3 ⊢ (ordTop‘ ≤ ) = (topGen‘((ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ∪ ran (,)))
5	4	eleq2i 2828	. 2 ⊢ (𝐴 ∈ (ordTop‘ ≤ ) ↔ 𝐴 ∈ (topGen‘((ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ∪ ran (,))))
6		tg2 22930	. . 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 4093	. . . . 5 ⊢ (𝑢 ∈ ((ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ∪ ran (,)) ↔ (𝑢 ∈ (ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ∨ 𝑢 ∈ ran (,)))
8		elun 4093	. . . . . . 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 5913	. . . . . . . . . 10 ⊢ (𝑢 ∈ V → (𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ↔ ∃𝑦 ∈ ℝ^* 𝑢 = (𝑦(,]+∞)))
11	10	elv 3434	. . . . . . . . 9 ⊢ (𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ↔ ∃𝑦 ∈ ℝ^* 𝑢 = (𝑦(,]+∞))
12		nltmnf 13080	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ^* → ¬ 𝑦 < -∞)
13		pnfxr 11199	. . . . . . . . . . . . . . . 16 ⊢ +∞ ∈ ℝ^*
14		elioc1 13340	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℝ^* ∧ +∞ ∈ ℝ^) → (-∞ ∈ (𝑦(,]+∞) ↔ (-∞ ∈ ℝ^ ∧ 𝑦 < -∞ ∧ -∞ ≤ +∞)))
15	13, 14	mpan2 692	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℝ^* → (-∞ ∈ (𝑦(,]+∞) ↔ (-∞ ∈ ℝ^* ∧ 𝑦 < -∞ ∧ -∞ ≤ +∞)))
16		simp2 1138	. . . . . . . . . . . . . . 15 ⊢ ((-∞ ∈ ℝ^* ∧ 𝑦 < -∞ ∧ -∞ ≤ +∞) → 𝑦 < -∞)
17	15, 16	biimtrdi 253	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ^* → (-∞ ∈ (𝑦(,]+∞) → 𝑦 < -∞))
18	12, 17	mtod 198	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ^* → ¬ -∞ ∈ (𝑦(,]+∞))
19		eleq2 2825	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (𝑦(,]+∞) → (-∞ ∈ 𝑢 ↔ -∞ ∈ (𝑦(,]+∞)))
20	19	notbid 318	. . . . . . . . . . . . 13 ⊢ (𝑢 = (𝑦(,]+∞) → (¬ -∞ ∈ 𝑢 ↔ ¬ -∞ ∈ (𝑦(,]+∞)))
21	18, 20	syl5ibrcom 247	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ^* → (𝑢 = (𝑦(,]+∞) → ¬ -∞ ∈ 𝑢))
22	21	rexlimiv 3131	. . . . . . . . . . 11 ⊢ (∃𝑦 ∈ ℝ^* 𝑢 = (𝑦(,]+∞) → ¬ -∞ ∈ 𝑢)
23	22	pm2.21d 121	. . . . . . . . . 10 ⊢ (∃𝑦 ∈ ℝ^* 𝑢 = (𝑦(,]+∞) → (-∞ ∈ 𝑢 → ∃𝑥 ∈ ℝ (-∞[,)𝑥) ⊆ 𝐴))
24	23	adantrd 491	. . . . . . . . 9 ⊢ (∃𝑦 ∈ ℝ^* 𝑢 = (𝑦(,]+∞) → ((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (-∞[,)𝑥) ⊆ 𝐴))
25	11, 24	sylbi 217	. . . . . . . 8 ⊢ (𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) → ((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (-∞[,)𝑥) ⊆ 𝐴))
26		eqid 2736	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)) = (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))
27	26	elrnmpt 5913	. . . . . . . . . 10 ⊢ (𝑢 ∈ V → (𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)) ↔ ∃𝑦 ∈ ℝ^* 𝑢 = (-∞[,)𝑦)))
28	27	elv 3434	. . . . . . . . 9 ⊢ (𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)) ↔ ∃𝑦 ∈ ℝ^* 𝑢 = (-∞[,)𝑦))
29		mnfxr 11202	. . . . . . . . . . . . . 14 ⊢ -∞ ∈ ℝ^*
30	29	a1i 11	. . . . . . . . . . . . 13 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → -∞ ∈ ℝ^*)
31		0xr 11192	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ^*
32		simprl 771	. . . . . . . . . . . . . 14 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → 𝑦 ∈ ℝ^*)
33		ifcl 4512	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → if(0 ≤ 𝑦, 0, 𝑦) ∈ ℝ^)
34	31, 32, 33	sylancr 588	. . . . . . . . . . . . 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 13076	. . . . . . . . . . . . . 14 ⊢ -∞ < 0
37		simpll 767	. . . . . . . . . . . . . . . . 17 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → -∞ ∈ 𝑢)
38		simprr 773	. . . . . . . . . . . . . . . . 17 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → 𝑢 = (-∞[,)𝑦))
39	37, 38	eleqtrd 2838	. . . . . . . . . . . . . . . 16 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → -∞ ∈ (-∞[,)𝑦))
40		elico1 13341	. . . . . . . . . . . . . . . . 17 ⊢ ((-∞ ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → (-∞ ∈ (-∞[,)𝑦) ↔ (-∞ ∈ ℝ^ ∧ -∞ ≤ -∞ ∧ -∞ < 𝑦)))
41	29, 32, 40	sylancr 588	. . . . . . . . . . . . . . . 16 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → (-∞ ∈ (-∞[,)𝑦) ↔ (-∞ ∈ ℝ^* ∧ -∞ ≤ -∞ ∧ -∞ < 𝑦)))
42	39, 41	mpbid 232	. . . . . . . . . . . . . . 15 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → (-∞ ∈ ℝ^* ∧ -∞ ≤ -∞ ∧ -∞ < 𝑦))
43	42	simp3d 1145	. . . . . . . . . . . . . 14 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → -∞ < 𝑦)
44		breq2 5089	. . . . . . . . . . . . . . 15 ⊢ (0 = if(0 ≤ 𝑦, 0, 𝑦) → (-∞ < 0 ↔ -∞ < if(0 ≤ 𝑦, 0, 𝑦)))
45		breq2 5089	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = if(0 ≤ 𝑦, 0, 𝑦) → (-∞ < 𝑦 ↔ -∞ < if(0 ≤ 𝑦, 0, 𝑦)))
46	44, 45	ifboth 4506	. . . . . . . . . . . . . 14 ⊢ ((-∞ < 0 ∧ -∞ < 𝑦) → -∞ < if(0 ≤ 𝑦, 0, 𝑦))
47	36, 43, 46	sylancr 588	. . . . . . . . . . . . 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 13129	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → if(0 ≤ 𝑦, 0, 𝑦) ≤ 0)
50	31, 32, 49	sylancr 588	. . . . . . . . . . . . . 14 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → if(0 ≤ 𝑦, 0, 𝑦) ≤ 0)
51		0re 11146	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℝ
52		ltpnf 13071	. . . . . . . . . . . . . . 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 13111	. . . . . . . . . . . . 13 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → if(0 ≤ 𝑦, 0, 𝑦) < +∞)
55		xrre2 13122	. . . . . . . . . . . . 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 1381	. . . . . . . . . . . 12 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → if(0 ≤ 𝑦, 0, 𝑦) ∈ ℝ)
57		xrmin2 13130	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → if(0 ≤ 𝑦, 0, 𝑦) ≤ 𝑦)
58	31, 32, 57	sylancr 588	. . . . . . . . . . . . . 14 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → if(0 ≤ 𝑦, 0, 𝑦) ≤ 𝑦)
59		df-ico 13304	. . . . . . . . . . . . . . 15 ⊢ [,) = (𝑎 ∈ ℝ^, 𝑏 ∈ ℝ^ ↦ {𝑐 ∈ ℝ^* ∣ (𝑎 ≤ 𝑐 ∧ 𝑐 < 𝑏)})
60		xrltletr 13108	. . . . . . . . . . . . . . 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 13317	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℝ^* ∧ if(0 ≤ 𝑦, 0, 𝑦) ≤ 𝑦) → (-∞[,)if(0 ≤ 𝑦, 0, 𝑦)) ⊆ (-∞[,)𝑦))
62	32, 58, 61	syl2anc 585	. . . . . . . . . . . . 13 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → (-∞[,)if(0 ≤ 𝑦, 0, 𝑦)) ⊆ (-∞[,)𝑦))
63		simplr 769	. . . . . . . . . . . . . 14 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → 𝑢 ⊆ 𝐴)
64	38, 63	eqsstrrd 3957	. . . . . . . . . . . . 13 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → (-∞[,)𝑦) ⊆ 𝐴)
65	62, 64	sstrd 3932	. . . . . . . . . . . 12 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → (-∞[,)if(0 ≤ 𝑦, 0, 𝑦)) ⊆ 𝐴)
66		oveq2 7375	. . . . . . . . . . . . . 14 ⊢ (𝑥 = if(0 ≤ 𝑦, 0, 𝑦) → (-∞[,)𝑥) = (-∞[,)if(0 ≤ 𝑦, 0, 𝑦)))
67	66	sseq1d 3953	. . . . . . . . . . . . 13 ⊢ (𝑥 = if(0 ≤ 𝑦, 0, 𝑦) → ((-∞[,)𝑥) ⊆ 𝐴 ↔ (-∞[,)if(0 ≤ 𝑦, 0, 𝑦)) ⊆ 𝐴))
68	67	rspcev 3564	. . . . . . . . . . . 12 ⊢ ((if(0 ≤ 𝑦, 0, 𝑦) ∈ ℝ ∧ (-∞[,)if(0 ≤ 𝑦, 0, 𝑦)) ⊆ 𝐴) → ∃𝑥 ∈ ℝ (-∞[,)𝑥) ⊆ 𝐴)
69	56, 65, 68	syl2anc 585	. . . . . . . . . . 11 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → ∃𝑥 ∈ ℝ (-∞[,)𝑥) ⊆ 𝐴)
70	69	rexlimdvaa 3139	. . . . . . . . . 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 217	. . . . . . . 8 ⊢ (𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)) → ((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (-∞[,)𝑥) ⊆ 𝐴))
73	25, 72	jaoi 858	. . . . . . 7 ⊢ ((𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∨ 𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) → ((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (-∞[,)𝑥) ⊆ 𝐴))
74	8, 73	sylbi 217	. . . . . 6 ⊢ (𝑢 ∈ (ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) → ((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (-∞[,)𝑥) ⊆ 𝐴))
75		mnfnre 11188	. . . . . . . . . 10 ⊢ -∞ ∉ ℝ
76	75	neli 3038	. . . . . . . . 9 ⊢ ¬ -∞ ∈ ℝ
77		elssuni 4881	. . . . . . . . . . 11 ⊢ (𝑢 ∈ ran (,) → 𝑢 ⊆ ∪ ran (,))
78		unirnioo 13402	. . . . . . . . . . 11 ⊢ ℝ = ∪ ran (,)
79	77, 78	sseqtrrdi 3963	. . . . . . . . . 10 ⊢ (𝑢 ∈ ran (,) → 𝑢 ⊆ ℝ)
80	79	sseld 3920	. . . . . . . . 9 ⊢ (𝑢 ∈ ran (,) → (-∞ ∈ 𝑢 → -∞ ∈ ℝ))
81	76, 80	mtoi 199	. . . . . . . 8 ⊢ (𝑢 ∈ ran (,) → ¬ -∞ ∈ 𝑢)
82	81	pm2.21d 121	. . . . . . 7 ⊢ (𝑢 ∈ ran (,) → (-∞ ∈ 𝑢 → ∃𝑥 ∈ ℝ (-∞[,)𝑥) ⊆ 𝐴))
83	82	adantrd 491	. . . . . 6 ⊢ (𝑢 ∈ ran (,) → ((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (-∞[,)𝑥) ⊆ 𝐴))
84	74, 83	jaoi 858	. . . . 5 ⊢ ((𝑢 ∈ (ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ∨ 𝑢 ∈ ran (,)) → ((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (-∞[,)𝑥) ⊆ 𝐴))
85	7, 84	sylbi 217	. . . 4 ⊢ (𝑢 ∈ ((ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ∪ ran (,)) → ((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (-∞[,)𝑥) ⊆ 𝐴))
86	85	rexlimiv 3131	. . 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 582	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 848 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∃wrex 3061 Vcvv 3429 ∪ cun 3887 ⊆ wss 3889 ifcif 4466 ∪ cuni 4850 class class class wbr 5085 ↦ cmpt 5166 ran crn 5632 ‘cfv 6498 (class class class)co 7367 ℝcr 11037 0cc0 11038 +∞cpnf 11176 -∞cmnf 11177 ℝ^*cxr 11178 < clt 11179 ≤ cle 11180 (,)cioo 13298 (,]cioc 13299 [,)cico 13300 topGenctg 17400 ordTopcordt 17463

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 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 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-1o 8405 df-2o 8406 df-er 8643 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fi 9324 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-ioo 13302 df-ioc 13303 df-ico 13304 df-icc 13305 df-topgen 17406 df-ordt 17465 df-ps 18532 df-tsr 18533 df-top 22859 df-bases 22911

This theorem is referenced by: xlimmnfvlem2 46261

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