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Theorem mnfnei 23159

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 2735	. . . 4 ⊢ ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) = ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞))
2		eqid 2735	. . . 4 ⊢ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)) = ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))
3		eqid 2735	. . . 4 ⊢ ran (,) = ran (,)
4	1, 2, 3	leordtval 23151	. . 3 ⊢ (ordTop‘ ≤ ) = (topGen‘((ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ∪ ran (,)))
5	4	eleq2i 2826	. 2 ⊢ (𝐴 ∈ (ordTop‘ ≤ ) ↔ 𝐴 ∈ (topGen‘((ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ∪ ran (,))))
6		tg2 22903	. . 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 4128	. . . . 5 ⊢ (𝑢 ∈ ((ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ∪ ran (,)) ↔ (𝑢 ∈ (ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ∨ 𝑢 ∈ ran (,)))
8		elun 4128	. . . . . . 7 ⊢ (𝑢 ∈ (ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ↔ (𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∨ 𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))))
9		eqid 2735	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) = (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞))
10	9	elrnmpt 5938	. . . . . . . . . 10 ⊢ (𝑢 ∈ V → (𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ↔ ∃𝑦 ∈ ℝ^* 𝑢 = (𝑦(,]+∞)))
11	10	elv 3464	. . . . . . . . 9 ⊢ (𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ↔ ∃𝑦 ∈ ℝ^* 𝑢 = (𝑦(,]+∞))
12		nltmnf 13145	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ^* → ¬ 𝑦 < -∞)
13		pnfxr 11289	. . . . . . . . . . . . . . . 16 ⊢ +∞ ∈ ℝ^*
14		elioc1 13404	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℝ^* ∧ +∞ ∈ ℝ^) → (-∞ ∈ (𝑦(,]+∞) ↔ (-∞ ∈ ℝ^ ∧ 𝑦 < -∞ ∧ -∞ ≤ +∞)))
15	13, 14	mpan2 691	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℝ^* → (-∞ ∈ (𝑦(,]+∞) ↔ (-∞ ∈ ℝ^* ∧ 𝑦 < -∞ ∧ -∞ ≤ +∞)))
16		simp2 1137	. . . . . . . . . . . . . . 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 2823	. . . . . . . . . . . . . 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 3134	. . . . . . . . . . 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 2735	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)) = (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))
27	26	elrnmpt 5938	. . . . . . . . . 10 ⊢ (𝑢 ∈ V → (𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)) ↔ ∃𝑦 ∈ ℝ^* 𝑢 = (-∞[,)𝑦)))
28	27	elv 3464	. . . . . . . . 9 ⊢ (𝑢 ∈ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)) ↔ ∃𝑦 ∈ ℝ^* 𝑢 = (-∞[,)𝑦))
29		mnfxr 11292	. . . . . . . . . . . . . 14 ⊢ -∞ ∈ ℝ^*
30	29	a1i 11	. . . . . . . . . . . . 13 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → -∞ ∈ ℝ^*)
31		0xr 11282	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ^*
32		simprl 770	. . . . . . . . . . . . . 14 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → 𝑦 ∈ ℝ^*)
33		ifcl 4546	. . . . . . . . . . . . . 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 13141	. . . . . . . . . . . . . 14 ⊢ -∞ < 0
37		simpll 766	. . . . . . . . . . . . . . . . 17 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → -∞ ∈ 𝑢)
38		simprr 772	. . . . . . . . . . . . . . . . 17 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → 𝑢 = (-∞[,)𝑦))
39	37, 38	eleqtrd 2836	. . . . . . . . . . . . . . . 16 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → -∞ ∈ (-∞[,)𝑦))
40		elico1 13405	. . . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . . 15 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → (-∞ ∈ ℝ^* ∧ -∞ ≤ -∞ ∧ -∞ < 𝑦))
43	42	simp3d 1144	. . . . . . . . . . . . . 14 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → -∞ < 𝑦)
44		breq2 5123	. . . . . . . . . . . . . . 15 ⊢ (0 = if(0 ≤ 𝑦, 0, 𝑦) → (-∞ < 0 ↔ -∞ < if(0 ≤ 𝑦, 0, 𝑦)))
45		breq2 5123	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = if(0 ≤ 𝑦, 0, 𝑦) → (-∞ < 𝑦 ↔ -∞ < if(0 ≤ 𝑦, 0, 𝑦)))
46	44, 45	ifboth 4540	. . . . . . . . . . . . . 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 13193	. . . . . . . . . . . . . . 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 11237	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℝ
52		ltpnf 13136	. . . . . . . . . . . . . . 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 13176	. . . . . . . . . . . . 13 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → if(0 ≤ 𝑦, 0, 𝑦) < +∞)
55		xrre2 13186	. . . . . . . . . . . . 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 1380	. . . . . . . . . . . 12 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → if(0 ≤ 𝑦, 0, 𝑦) ∈ ℝ)
57		xrmin2 13194	. . . . . . . . . . . . . . 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 13368	. . . . . . . . . . . . . . 15 ⊢ [,) = (𝑎 ∈ ℝ^, 𝑏 ∈ ℝ^ ↦ {𝑐 ∈ ℝ^* ∣ (𝑎 ≤ 𝑐 ∧ 𝑐 < 𝑏)})
60		xrltletr 13173	. . . . . . . . . . . . . . 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 13381	. . . . . . . . . . . . . 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 768	. . . . . . . . . . . . . 14 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → 𝑢 ⊆ 𝐴)
64	38, 63	eqsstrrd 3994	. . . . . . . . . . . . 13 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → (-∞[,)𝑦) ⊆ 𝐴)
65	62, 64	sstrd 3969	. . . . . . . . . . . 12 ⊢ (((-∞ ∈ 𝑢 ∧ 𝑢 ⊆ 𝐴) ∧ (𝑦 ∈ ℝ^* ∧ 𝑢 = (-∞[,)𝑦))) → (-∞[,)if(0 ≤ 𝑦, 0, 𝑦)) ⊆ 𝐴)
66		oveq2 7413	. . . . . . . . . . . . . 14 ⊢ (𝑥 = if(0 ≤ 𝑦, 0, 𝑦) → (-∞[,)𝑥) = (-∞[,)if(0 ≤ 𝑦, 0, 𝑦)))
67	66	sseq1d 3990	. . . . . . . . . . . . 13 ⊢ (𝑥 = if(0 ≤ 𝑦, 0, 𝑦) → ((-∞[,)𝑥) ⊆ 𝐴 ↔ (-∞[,)if(0 ≤ 𝑦, 0, 𝑦)) ⊆ 𝐴))
68	67	rspcev 3601	. . . . . . . . . . . 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 3142	. . . . . . . . . 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 857	. . . . . . 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 11278	. . . . . . . . . 10 ⊢ -∞ ∉ ℝ
76	75	neli 3038	. . . . . . . . 9 ⊢ ¬ -∞ ∈ ℝ
77		elssuni 4913	. . . . . . . . . . 11 ⊢ (𝑢 ∈ ran (,) → 𝑢 ⊆ ∪ ran (,))
78		unirnioo 13466	. . . . . . . . . . 11 ⊢ ℝ = ∪ ran (,)
79	77, 78	sseqtrrdi 4000	. . . . . . . . . 10 ⊢ (𝑢 ∈ ran (,) → 𝑢 ⊆ ℝ)
80	79	sseld 3957	. . . . . . . . 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 857	. . . . 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 3134	. . 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 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1540 ∈ wcel 2108 ∃wrex 3060 Vcvv 3459 ∪ cun 3924 ⊆ wss 3926 ifcif 4500 ∪ cuni 4883 class class class wbr 5119 ↦ cmpt 5201 ran crn 5655 ‘cfv 6531 (class class class)co 7405 ℝcr 11128 0cc0 11129 +∞cpnf 11266 -∞cmnf 11267 ℝ^*cxr 11268 < clt 11269 ≤ cle 11270 (,)cioo 13362 (,]cioc 13363 [,)cico 13364 topGenctg 17451 ordTopcordt 17513

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206

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 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7862 df-1st 7988 df-2nd 7989 df-1o 8480 df-2o 8481 df-er 8719 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-fi 9423 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-ioo 13366 df-ioc 13367 df-ico 13368 df-icc 13369 df-topgen 17457 df-ordt 17515 df-ps 18576 df-tsr 18577 df-top 22832 df-bases 22884

This theorem is referenced by: xlimmnfvlem2 45862

W3C validator