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Theorem mnfnei 23545

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 2760	. . . 4 ⊢ ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) = ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞))
2		eqid 2760	. . . 4 ⊢ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)) = ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))
3		eqid 2760	. . . 4 ⊢ ran (,) = ran (,)
4	1, 2, 3	leordtval 23537	. . 3 ⊢ (ordTop‘ ≤ ) = (topGen‘((ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ∪ ran (,)))
5	4	eleq2i 2852	. 2 ⊢ (𝐴 ∈ (ordTop‘ ≤ ) ↔ 𝐴 ∈ (topGen‘((ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ∪ ran (,))))
6		tg2 23289	. . 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 4152	. . . . 5 ⊢ (𝑢 ∈ ((ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ∪ ran (,)) ↔ (𝑢 ∈ (ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ∨ 𝑢 ∈ ran (,)))
8		elun 4152	. . . . . . 7 ⊢ (𝑢 ∈ (ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ↔ (𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∨ 𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))))
9		eqid 2760	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) = (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞))
10	9	elrnmpt 5939	. . . . . . . . . 10 ⊢ (𝑢 ∈ V → (𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ↔ ∃𝑦 ∈ ℝ^* 𝑢 = (𝑦(,]+∞)))
11	10	elv 3455	. . . . . . . . 9 ⊢ (𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ↔ ∃𝑦 ∈ ℝ^* 𝑢 = (𝑦(,]+∞))
12		nltmnf 13262	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ^* → ¬ 𝑦 < -∞)
13		pnfxr 11367	. . . . . . . . . . . . . . . 16 ⊢ +∞ ∈ ℝ^*
14		elioc1 13522	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℝ^* ∧ +∞ ∈ ℝ^) → (-∞ ∈ (𝑦(,]+∞) ↔ (-∞ ∈ ℝ^ ∧ 𝑦 < -∞ ∧ -∞ ≤ +∞)))
15	13, 14	mpan2 704	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℝ^* → (-∞ ∈ (𝑦(,]+∞) ↔ (-∞ ∈ ℝ^* ∧ 𝑦 < -∞ ∧ -∞ ≤ +∞)))
16		simp2 1155	. . . . . . . . . . . . . . 15 ⊢ ((-∞ ∈ ℝ^* ∧ 𝑦 < -∞ ∧ -∞ ≤ +∞) → 𝑦 < -∞)
17	15, 16	biimtrdi 256	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ^* → (-∞ ∈ (𝑦(,]+∞) → 𝑦 < -∞))
18	12, 17	mtod 201	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ^* → ¬ -∞ ∈ (𝑦(,]+∞))
19		eleq2 2849	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (𝑦(,]+∞) → (-∞ ∈ 𝑢 ↔ -∞ ∈ (𝑦(,]+∞)))
20	19	notbid 321	. . . . . . . . . . . . 13 ⊢ (𝑢 = (𝑦(,]+∞) → (¬ -∞ ∈ 𝑢 ↔ ¬ -∞ ∈ (𝑦(,]+∞)))
21	18, 20	syl5ibrcom 250	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ^* → (𝑢 = (𝑦(,]+∞) → ¬ -∞ ∈ 𝑢))
22	21	rexlimiv 3156	. . . . . . . . . . 11 ⊢ (∃𝑦 ∈ ℝ^* 𝑢 = (𝑦(,]+∞) → ¬ -∞ ∈ 𝑢)
23	22	pm2.21d 122	. . . . . . . . . 10 ⊢ (∃𝑦 ∈ ℝ^* 𝑢 = (𝑦(,]+∞) → (-∞ ∈ 𝑢 → ∃𝑥 ∈ ℝ (-∞[,)𝑥) ⊆ 𝐴))
24	23	adantrd 497	. . . . . . . . 9 ⊢ (∃𝑦 ∈ ℝ^* 𝑢 = (𝑦(,]+∞) → ((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (-∞[,)𝑥) ⊆ 𝐴))
25	11, 24	sylbi 220	. . . . . . . 8 ⊢ (𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) → ((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (-∞[,)𝑥) ⊆ 𝐴))
26		eqid 2760	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)) = (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))
27	26	elrnmpt 5939	. . . . . . . . . 10 ⊢ (𝑢 ∈ V → (𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)) ↔ ∃𝑦 ∈ ℝ^* 𝑢 = (-∞[,)𝑦)))
28	27	elv 3455	. . . . . . . . 9 ⊢ (𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)) ↔ ∃𝑦 ∈ ℝ^* 𝑢 = (-∞[,)𝑦))
29		mnfxr 11370	. . . . . . . . . . . . . 14 ⊢ -∞ ∈ ℝ^*
30	29	a1i 11	. . . . . . . . . . . . 13 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → -∞ ∈ ℝ^*)
31		0xr 11360	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ^*
32		simprl 783	. . . . . . . . . . . . . 14 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → 𝑦 ∈ ℝ^*)
33		ifcl 4527	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → if(0 ≤ 𝑦, 0, 𝑦) ∈ ℝ^)
34	31, 32, 33	sylancr 599	. . . . . . . . . . . . 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 13258	. . . . . . . . . . . . . 14 ⊢ -∞ < 0
37		simpll 779	. . . . . . . . . . . . . . . . 17 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → -∞ ∈ 𝑢)
38		simprr 785	. . . . . . . . . . . . . . . . 17 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → 𝑢 = (-∞[,)𝑦))
39	37, 38	eleqtrd 2862	. . . . . . . . . . . . . . . 16 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → -∞ ∈ (-∞[,)𝑦))
40		elico1 13523	. . . . . . . . . . . . . . . . 17 ⊢ ((-∞ ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → (-∞ ∈ (-∞[,)𝑦) ↔ (-∞ ∈ ℝ^ ∧ -∞ ≤ -∞ ∧ -∞ < 𝑦)))
41	29, 32, 40	sylancr 599	. . . . . . . . . . . . . . . 16 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → (-∞ ∈ (-∞[,)𝑦) ↔ (-∞ ∈ ℝ^* ∧ -∞ ≤ -∞ ∧ -∞ < 𝑦)))
42	39, 41	mpbid 235	. . . . . . . . . . . . . . 15 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → (-∞ ∈ ℝ^* ∧ -∞ ≤ -∞ ∧ -∞ < 𝑦))
43	42	simp3d 1162	. . . . . . . . . . . . . 14 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → -∞ < 𝑦)
44		breq2 5106	. . . . . . . . . . . . . . 15 ⊢ (0 = if(0 ≤ 𝑦, 0, 𝑦) → (-∞ < 0 ↔ -∞ < if(0 ≤ 𝑦, 0, 𝑦)))
45		breq2 5106	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = if(0 ≤ 𝑦, 0, 𝑦) → (-∞ < 𝑦 ↔ -∞ < if(0 ≤ 𝑦, 0, 𝑦)))
46	44, 45	ifboth 4521	. . . . . . . . . . . . . 14 ⊢ ((-∞ < 0 ∧ -∞ < 𝑦) → -∞ < if(0 ≤ 𝑦, 0, 𝑦))
47	36, 43, 46	sylancr 599	. . . . . . . . . . . . 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 13311	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → if(0 ≤ 𝑦, 0, 𝑦) ≤ 0)
50	31, 32, 49	sylancr 599	. . . . . . . . . . . . . 14 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → if(0 ≤ 𝑦, 0, 𝑦) ≤ 0)
51		0re 11314	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℝ
52		ltpnf 13253	. . . . . . . . . . . . . . 15 ⊢ (0 ∈ ℝ → 0 < +∞)
53	51, 52	mp1i 14	. . . . . . . . . . . . . 14 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → 0 < +∞)
54	34, 48, 35, 50, 53	xrlelttrd 13293	. . . . . . . . . . . . 13 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → if(0 ≤ 𝑦, 0, 𝑦) < +∞)
55		xrre2 13304	. . . . . . . . . . . . 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 1405	. . . . . . . . . . . 12 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → if(0 ≤ 𝑦, 0, 𝑦) ∈ ℝ)
57		xrmin2 13312	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → if(0 ≤ 𝑦, 0, 𝑦) ≤ 𝑦)
58	31, 32, 57	sylancr 599	. . . . . . . . . . . . . 14 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → if(0 ≤ 𝑦, 0, 𝑦) ≤ 𝑦)
59		df-ico 13486	. . . . . . . . . . . . . . 15 ⊢ [,) = (𝑎 ∈ ℝ^, 𝑏 ∈ ℝ^ ↦ {𝑐 ∈ ℝ^* ∣ (𝑎 ≤ 𝑐 ∧ 𝑐 < 𝑏)})
60		xrltletr 13290	. . . . . . . . . . . . . . 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 13499	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℝ^* ∧ if(0 ≤ 𝑦, 0, 𝑦) ≤ 𝑦) → (-∞[,)if(0 ≤ 𝑦, 0, 𝑦)) ⊆ (-∞[,)𝑦))
62	32, 58, 61	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → (-∞[,)if(0 ≤ 𝑦, 0, 𝑦)) ⊆ (-∞[,)𝑦))
63		simplr 781	. . . . . . . . . . . . . 14 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → 𝑢 ⊆ 𝐴)
64	38, 63	eqsstrrd 3943	. . . . . . . . . . . . 13 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → (-∞[,)𝑦) ⊆ 𝐴)
65	62, 64	sstrd 3918	. . . . . . . . . . . 12 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → (-∞[,)if(0 ≤ 𝑦, 0, 𝑦)) ⊆ 𝐴)
66		oveq2 7429	. . . . . . . . . . . . . 14 ⊢ (𝑥 = if(0 ≤ 𝑦, 0, 𝑦) → (-∞[,)𝑥) = (-∞[,)if(0 ≤ 𝑦, 0, 𝑦)))
67	66	sseq1d 3939	. . . . . . . . . . . . 13 ⊢ (𝑥 = if(0 ≤ 𝑦, 0, 𝑦) → ((-∞[,)𝑥) ⊆ 𝐴 ↔ (-∞[,)if(0 ≤ 𝑦, 0, 𝑦)) ⊆ 𝐴))
68	67	rspcev 3576	. . . . . . . . . . . 12 ⊢ ((if(0 ≤ 𝑦, 0, 𝑦) ∈ ℝ ∧ (-∞[,)if(0 ≤ 𝑦, 0, 𝑦)) ⊆ 𝐴) → ∃𝑥 ∈ ℝ (-∞[,)𝑥) ⊆ 𝐴)
69	56, 65, 68	syl2anc 596	. . . . . . . . . . 11 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → ∃𝑥 ∈ ℝ (-∞[,)𝑥) ⊆ 𝐴)
70	69	rexlimdvaa 3164	. . . . . . . . . 10 ⊢ ((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → (∃𝑦 ∈ ℝ^* 𝑢 = (-∞[,)𝑦) → ∃𝑥 ∈ ℝ (-∞[,)𝑥) ⊆ 𝐴))
71	70	com12 33	. . . . . . . . 9 ⊢ (∃𝑦 ∈ ℝ^* 𝑢 = (-∞[,)𝑦) → ((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (-∞[,)𝑥) ⊆ 𝐴))
72	28, 71	sylbi 220	. . . . . . . 8 ⊢ (𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)) → ((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (-∞[,)𝑥) ⊆ 𝐴))
73	25, 72	jaoi 871	. . . . . . 7 ⊢ ((𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∨ 𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) → ((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (-∞[,)𝑥) ⊆ 𝐴))
74	8, 73	sylbi 220	. . . . . 6 ⊢ (𝑢 ∈ (ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) → ((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (-∞[,)𝑥) ⊆ 𝐴))
75		mnfnre 11356	. . . . . . . . . 10 ⊢ -∞ ∉ ℝ
76	75	neli 3063	. . . . . . . . 9 ⊢ ¬ -∞ ∈ ℝ
77		elssuni 4898	. . . . . . . . . . 11 ⊢ (𝑢 ∈ ran (,) → 𝑢 ⊆ ∪ ran (,))
78		unirnioo 13584	. . . . . . . . . . 11 ⊢ ℝ = ∪ ran (,)
79	77, 78	sseqtrrdi 3949	. . . . . . . . . 10 ⊢ (𝑢 ∈ ran (,) → 𝑢 ⊆ ℝ)
80	79	sseld 3907	. . . . . . . . 9 ⊢ (𝑢 ∈ ran (,) → (-∞ ∈ 𝑢 → -∞ ∈ ℝ))
81	76, 80	mtoi 202	. . . . . . . 8 ⊢ (𝑢 ∈ ran (,) → ¬ -∞ ∈ 𝑢)
82	81	pm2.21d 122	. . . . . . 7 ⊢ (𝑢 ∈ ran (,) → (-∞ ∈ 𝑢 → ∃𝑥 ∈ ℝ (-∞[,)𝑥) ⊆ 𝐴))
83	82	adantrd 497	. . . . . 6 ⊢ (𝑢 ∈ ran (,) → ((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (-∞[,)𝑥) ⊆ 𝐴))
84	74, 83	jaoi 871	. . . . 5 ⊢ ((𝑢 ∈ (ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ∨ 𝑢 ∈ ran (,)) → ((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (-∞[,)𝑥) ⊆ 𝐴))
85	7, 84	sylbi 220	. . . 4 ⊢ (𝑢 ∈ ((ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ∪ ran (,)) → ((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (-∞[,)𝑥) ⊆ 𝐴))
86	85	rexlimiv 3156	. . 3 ⊢ (∃𝑢 ∈ ((ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ∪ ran (,))(-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) → ∃𝑥 ∈ ℝ (-∞[,)𝑥) ⊆ 𝐴)
87	6, 86	syl 18	. 2 ⊢ ((𝐴 ∈ (topGen‘((ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ∪ ran (,))) ∧ -∞ ∈ 𝐴) → ∃𝑥 ∈ ℝ (-∞[,)𝑥) ⊆ 𝐴)
88	5, 87	sylanb 593	1 ⊢ ((𝐴 ∈ (ordTop‘ ≤ ) ∧ -∞ ∈ 𝐴) → ∃𝑥 ∈ ℝ (-∞[,)𝑥) ⊆ 𝐴)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ∃wrex 3086 Vcvv 3450 ⊆ wss 3895 ∪ cun 4149 ifcif 4481 ∪ cuni 4866 class class class wbr 5102 ↦ cmpt 5185 ran crn 5651 ‘cfv 6539 (class class class)co 7421 ℝcr 11203 0cc0 11204 +∞cpnf 11344 -∞cmnf 11345 ℝ^*cxr 11346 < clt 11347 ≤ cle 11348 (,)cioo 13480 (,]cioc 13481 [,)cico 13482 topGenctg 17614 ordTopcordt 17677

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7752 ax-cnex 11260 ax-resscn 11261 ax-1cn 11262 ax-icn 11263 ax-addcl 11264 ax-addrcl 11265 ax-mulcl 11266 ax-mulrcl 11267 ax-mulcom 11268 ax-addass 11269 ax-mulass 11270 ax-distr 11271 ax-i2m1 11272 ax-1ne0 11273 ax-1rid 11274 ax-rnegex 11275 ax-rrecex 11276 ax-cnre 11277 ax-pre-lttri 11278 ax-pre-lttrn 11279 ax-pre-ltadd 11280 ax-pre-mulgt0 11281

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7378 df-ov 7424 df-oprab 7425 df-mpo 7426 df-om 7879 df-1st 8002 df-2nd 8003 df-1o 8480 df-2o 8481 df-er 8721 df-en 8978 df-dom 8979 df-sdom 8980 df-fin 8981 df-fi 9407 df-pnf 11349 df-mnf 11350 df-xr 11351 df-ltxr 11352 df-le 11353 df-sub 11547 df-neg 11548 df-ioo 13484 df-ioc 13485 df-ico 13486 df-icc 13487 df-topgen 17620 df-ordt 17679 df-ps 18746 df-tsr 18747 df-top 23218 df-bases 23270

This theorem is used by: xlimmnfvlem2 46937

W3C validator