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Theorem supinfneg 8683

Description: If a set of real numbers has a least upper bound, the set of the negation of those numbers has a greatest lower bound. For a theorem which is similar but only for the boundedness part, see ublbneg 8698. (Contributed by Jim Kingdon, 15-Jan-2022.)

**Hypotheses**
Ref	Expression
supinfneg.ex	⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
supinfneg.ss	⊢ (𝜑 → 𝐴 ⊆ ℝ)

**Assertion**
Ref	Expression
supinfneg	⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴}𝑧 < 𝑦)))

Distinct variable groups: 𝑦,𝐴,𝑧,𝑤,𝑥 𝜑,𝑦 Allowed substitution hints: 𝜑(𝑥,𝑧,𝑤)

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

Step	Hyp	Ref	Expression
1		supinfneg.ex	. . . 4 ⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
2		breq1 3788	. . . . . . . 8 ⊢ (𝑎 = 𝑥 → (𝑎 < 𝑦 ↔ 𝑥 < 𝑦))
3	2	notbid 624	. . . . . . 7 ⊢ (𝑎 = 𝑥 → (¬ 𝑎 < 𝑦 ↔ ¬ 𝑥 < 𝑦))
4	3	ralbidv 2368	. . . . . 6 ⊢ (𝑎 = 𝑥 → (∀𝑦 ∈ 𝐴 ¬ 𝑎 < 𝑦 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦))
5		breq2 3789	. . . . . . . 8 ⊢ (𝑎 = 𝑥 → (𝑦 < 𝑎 ↔ 𝑦 < 𝑥))
6	5	imbi1d 229	. . . . . . 7 ⊢ (𝑎 = 𝑥 → ((𝑦 < 𝑎 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧) ↔ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
7	6	ralbidv 2368	. . . . . 6 ⊢ (𝑎 = 𝑥 → (∀𝑦 ∈ ℝ (𝑦 < 𝑎 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧) ↔ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
8	4, 7	anbi12d 456	. . . . 5 ⊢ (𝑎 = 𝑥 → ((∀𝑦 ∈ 𝐴 ¬ 𝑎 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑎 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)) ↔ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))))
9	8	cbvrexv 2578	. . . 4 ⊢ (∃𝑎 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑎 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑎 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)) ↔ ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
10	1, 9	sylibr 132	. . 3 ⊢ (𝜑 → ∃𝑎 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑎 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑎 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
11		breq2 3789	. . . . . . 7 ⊢ (𝑏 = 𝑦 → (𝑎 < 𝑏 ↔ 𝑎 < 𝑦))
12	11	notbid 624	. . . . . 6 ⊢ (𝑏 = 𝑦 → (¬ 𝑎 < 𝑏 ↔ ¬ 𝑎 < 𝑦))
13	12	cbvralv 2577	. . . . 5 ⊢ (∀𝑏 ∈ 𝐴 ¬ 𝑎 < 𝑏 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑎 < 𝑦)
14		breq2 3789	. . . . . . . . 9 ⊢ (𝑐 = 𝑧 → (𝑏 < 𝑐 ↔ 𝑏 < 𝑧))
15	14	cbvrexv 2578	. . . . . . . 8 ⊢ (∃𝑐 ∈ 𝐴 𝑏 < 𝑐 ↔ ∃𝑧 ∈ 𝐴 𝑏 < 𝑧)
16	15	imbi2i 224	. . . . . . 7 ⊢ ((𝑏 < 𝑎 → ∃𝑐 ∈ 𝐴 𝑏 < 𝑐) ↔ (𝑏 < 𝑎 → ∃𝑧 ∈ 𝐴 𝑏 < 𝑧))
17	16	ralbii 2372	. . . . . 6 ⊢ (∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐 ∈ 𝐴 𝑏 < 𝑐) ↔ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑧 ∈ 𝐴 𝑏 < 𝑧))
18		breq1 3788	. . . . . . . 8 ⊢ (𝑏 = 𝑦 → (𝑏 < 𝑎 ↔ 𝑦 < 𝑎))
19		breq1 3788	. . . . . . . . 9 ⊢ (𝑏 = 𝑦 → (𝑏 < 𝑧 ↔ 𝑦 < 𝑧))
20	19	rexbidv 2369	. . . . . . . 8 ⊢ (𝑏 = 𝑦 → (∃𝑧 ∈ 𝐴 𝑏 < 𝑧 ↔ ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))
21	18, 20	imbi12d 232	. . . . . . 7 ⊢ (𝑏 = 𝑦 → ((𝑏 < 𝑎 → ∃𝑧 ∈ 𝐴 𝑏 < 𝑧) ↔ (𝑦 < 𝑎 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
22	21	cbvralv 2577	. . . . . 6 ⊢ (∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑧 ∈ 𝐴 𝑏 < 𝑧) ↔ ∀𝑦 ∈ ℝ (𝑦 < 𝑎 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))
23	17, 22	bitri 182	. . . . 5 ⊢ (∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐 ∈ 𝐴 𝑏 < 𝑐) ↔ ∀𝑦 ∈ ℝ (𝑦 < 𝑎 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))
24	13, 23	anbi12i 447	. . . 4 ⊢ ((∀𝑏 ∈ 𝐴 ¬ 𝑎 < 𝑏 ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐 ∈ 𝐴 𝑏 < 𝑐)) ↔ (∀𝑦 ∈ 𝐴 ¬ 𝑎 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑎 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
25	24	rexbii 2373	. . 3 ⊢ (∃𝑎 ∈ ℝ (∀𝑏 ∈ 𝐴 ¬ 𝑎 < 𝑏 ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐 ∈ 𝐴 𝑏 < 𝑐)) ↔ ∃𝑎 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑎 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑎 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
26	10, 25	sylibr 132	. 2 ⊢ (𝜑 → ∃𝑎 ∈ ℝ (∀𝑏 ∈ 𝐴 ¬ 𝑎 < 𝑏 ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐 ∈ 𝐴 𝑏 < 𝑐)))
27		renegcl 7369	. . . . . 6 ⊢ (𝑎 ∈ ℝ → -𝑎 ∈ ℝ)
28	27	ad2antlr 472	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ (∀𝑏 ∈ 𝐴 ¬ 𝑎 < 𝑏 ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐 ∈ 𝐴 𝑏 < 𝑐))) → -𝑎 ∈ ℝ)
29		simplr 496	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ (∀𝑏 ∈ 𝐴 ¬ 𝑎 < 𝑏 ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐 ∈ 𝐴 𝑏 < 𝑐))) → 𝑎 ∈ ℝ)
30		simprl 497	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ (∀𝑏 ∈ 𝐴 ¬ 𝑎 < 𝑏 ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐 ∈ 𝐴 𝑏 < 𝑐))) → ∀𝑏 ∈ 𝐴 ¬ 𝑎 < 𝑏)
31		elrabi 2746	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} → 𝑦 ∈ ℝ)
32		negeq 7301	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑦 → -𝑤 = -𝑦)
33	32	eleq1d 2147	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑦 → (-𝑤 ∈ 𝐴 ↔ -𝑦 ∈ 𝐴))
34	33	elrab3 2750	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ → (𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} ↔ -𝑦 ∈ 𝐴))
35	34	biimpd 142	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ → (𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} → -𝑦 ∈ 𝐴))
36	31, 35	mpcom 36	. . . . . . . . . . 11 ⊢ (𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} → -𝑦 ∈ 𝐴)
37		breq2 3789	. . . . . . . . . . . . 13 ⊢ (𝑏 = -𝑦 → (𝑎 < 𝑏 ↔ 𝑎 < -𝑦))
38	37	notbid 624	. . . . . . . . . . . 12 ⊢ (𝑏 = -𝑦 → (¬ 𝑎 < 𝑏 ↔ ¬ 𝑎 < -𝑦))
39	38	rspcv 2697	. . . . . . . . . . 11 ⊢ (-𝑦 ∈ 𝐴 → (∀𝑏 ∈ 𝐴 ¬ 𝑎 < 𝑏 → ¬ 𝑎 < -𝑦))
40	36, 39	syl 14	. . . . . . . . . 10 ⊢ (𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} → (∀𝑏 ∈ 𝐴 ¬ 𝑎 < 𝑏 → ¬ 𝑎 < -𝑦))
41	40	adantr 270	. . . . . . . . 9 ⊢ ((𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} ∧ 𝑎 ∈ ℝ) → (∀𝑏 ∈ 𝐴 ¬ 𝑎 < 𝑏 → ¬ 𝑎 < -𝑦))
42		ltnegcon2 7568	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℝ ∧ 𝑎 ∈ ℝ) → (𝑦 < -𝑎 ↔ 𝑎 < -𝑦))
43	42	notbid 624	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℝ ∧ 𝑎 ∈ ℝ) → (¬ 𝑦 < -𝑎 ↔ ¬ 𝑎 < -𝑦))
44	31, 43	sylan 277	. . . . . . . . 9 ⊢ ((𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} ∧ 𝑎 ∈ ℝ) → (¬ 𝑦 < -𝑎 ↔ ¬ 𝑎 < -𝑦))
45	41, 44	sylibrd 167	. . . . . . . 8 ⊢ ((𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} ∧ 𝑎 ∈ ℝ) → (∀𝑏 ∈ 𝐴 ¬ 𝑎 < 𝑏 → ¬ 𝑦 < -𝑎))
46	45	ancoms 264	. . . . . . 7 ⊢ ((𝑎 ∈ ℝ ∧ 𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴}) → (∀𝑏 ∈ 𝐴 ¬ 𝑎 < 𝑏 → ¬ 𝑦 < -𝑎))
47	46	ralrimdva 2441	. . . . . 6 ⊢ (𝑎 ∈ ℝ → (∀𝑏 ∈ 𝐴 ¬ 𝑎 < 𝑏 → ∀𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} ¬ 𝑦 < -𝑎))
48	29, 30, 47	sylc 61	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ (∀𝑏 ∈ 𝐴 ¬ 𝑎 < 𝑏 ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐 ∈ 𝐴 𝑏 < 𝑐))) → ∀𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} ¬ 𝑦 < -𝑎)
49		nfv 1461	. . . . . . . . . . . 12 ⊢ Ⅎ𝑐(𝜑 ∧ 𝑎 ∈ ℝ)
50		nfcv 2219	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑐ℝ
51		nfv 1461	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑐 𝑏 < 𝑎
52		nfre1 2407	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑐∃𝑐 ∈ 𝐴 𝑏 < 𝑐
53	51, 52	nfim 1504	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑐(𝑏 < 𝑎 → ∃𝑐 ∈ 𝐴 𝑏 < 𝑐)
54	50, 53	nfralya 2404	. . . . . . . . . . . 12 ⊢ Ⅎ𝑐∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐 ∈ 𝐴 𝑏 < 𝑐)
55	49, 54	nfan 1497	. . . . . . . . . . 11 ⊢ Ⅎ𝑐((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐 ∈ 𝐴 𝑏 < 𝑐))
56		nfv 1461	. . . . . . . . . . 11 ⊢ Ⅎ𝑐 𝑦 ∈ ℝ
57	55, 56	nfan 1497	. . . . . . . . . 10 ⊢ Ⅎ𝑐(((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐 ∈ 𝐴 𝑏 < 𝑐)) ∧ 𝑦 ∈ ℝ)
58		nfv 1461	. . . . . . . . . 10 ⊢ Ⅎ𝑐-𝑎 < 𝑦
59	57, 58	nfan 1497	. . . . . . . . 9 ⊢ Ⅎ𝑐((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐 ∈ 𝐴 𝑏 < 𝑐)) ∧ 𝑦 ∈ ℝ) ∧ -𝑎 < 𝑦)
60		simplr 496	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐 ∈ 𝐴 𝑏 < 𝑐)) ∧ 𝑦 ∈ ℝ) ∧ -𝑎 < 𝑦) ∧ 𝑐 ∈ 𝐴) ∧ -𝑦 < 𝑐) → 𝑐 ∈ 𝐴)
61		supinfneg.ss	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
62	61	sseld 2998	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑐 ∈ 𝐴 → 𝑐 ∈ ℝ))
63	62	ad6antr 481	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐 ∈ 𝐴 𝑏 < 𝑐)) ∧ 𝑦 ∈ ℝ) ∧ -𝑎 < 𝑦) ∧ 𝑐 ∈ 𝐴) ∧ -𝑦 < 𝑐) → (𝑐 ∈ 𝐴 → 𝑐 ∈ ℝ))
64	60, 63	mpd 13	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐 ∈ 𝐴 𝑏 < 𝑐)) ∧ 𝑦 ∈ ℝ) ∧ -𝑎 < 𝑦) ∧ 𝑐 ∈ 𝐴) ∧ -𝑦 < 𝑐) → 𝑐 ∈ ℝ)
65	64	renegcld 7484	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐 ∈ 𝐴 𝑏 < 𝑐)) ∧ 𝑦 ∈ ℝ) ∧ -𝑎 < 𝑦) ∧ 𝑐 ∈ 𝐴) ∧ -𝑦 < 𝑐) → -𝑐 ∈ ℝ)
66	64	recnd 7147	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐 ∈ 𝐴 𝑏 < 𝑐)) ∧ 𝑦 ∈ ℝ) ∧ -𝑎 < 𝑦) ∧ 𝑐 ∈ 𝐴) ∧ -𝑦 < 𝑐) → 𝑐 ∈ ℂ)
67	66	negnegd 7410	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐 ∈ 𝐴 𝑏 < 𝑐)) ∧ 𝑦 ∈ ℝ) ∧ -𝑎 < 𝑦) ∧ 𝑐 ∈ 𝐴) ∧ -𝑦 < 𝑐) → --𝑐 = 𝑐)
68	67, 60	eqeltrd 2155	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐 ∈ 𝐴 𝑏 < 𝑐)) ∧ 𝑦 ∈ ℝ) ∧ -𝑎 < 𝑦) ∧ 𝑐 ∈ 𝐴) ∧ -𝑦 < 𝑐) → --𝑐 ∈ 𝐴)
69		negeq 7301	. . . . . . . . . . . . 13 ⊢ (𝑤 = -𝑐 → -𝑤 = --𝑐)
70	69	eleq1d 2147	. . . . . . . . . . . 12 ⊢ (𝑤 = -𝑐 → (-𝑤 ∈ 𝐴 ↔ --𝑐 ∈ 𝐴))
71	70	elrab 2749	. . . . . . . . . . 11 ⊢ (-𝑐 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} ↔ (-𝑐 ∈ ℝ ∧ --𝑐 ∈ 𝐴))
72	65, 68, 71	sylanbrc 408	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐 ∈ 𝐴 𝑏 < 𝑐)) ∧ 𝑦 ∈ ℝ) ∧ -𝑎 < 𝑦) ∧ 𝑐 ∈ 𝐴) ∧ -𝑦 < 𝑐) → -𝑐 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴})
73		simp-4r 508	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐 ∈ 𝐴 𝑏 < 𝑐)) ∧ 𝑦 ∈ ℝ) ∧ -𝑎 < 𝑦) ∧ 𝑐 ∈ 𝐴) ∧ -𝑦 < 𝑐) → 𝑦 ∈ ℝ)
74		simpr 108	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐 ∈ 𝐴 𝑏 < 𝑐)) ∧ 𝑦 ∈ ℝ) ∧ -𝑎 < 𝑦) ∧ 𝑐 ∈ 𝐴) ∧ -𝑦 < 𝑐) → -𝑦 < 𝑐)
75	73, 64, 74	ltnegcon1d 7625	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐 ∈ 𝐴 𝑏 < 𝑐)) ∧ 𝑦 ∈ ℝ) ∧ -𝑎 < 𝑦) ∧ 𝑐 ∈ 𝐴) ∧ -𝑦 < 𝑐) → -𝑐 < 𝑦)
76		breq1 3788	. . . . . . . . . . 11 ⊢ (𝑧 = -𝑐 → (𝑧 < 𝑦 ↔ -𝑐 < 𝑦))
77	76	rspcev 2701	. . . . . . . . . 10 ⊢ ((-𝑐 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} ∧ -𝑐 < 𝑦) → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴}𝑧 < 𝑦)
78	72, 75, 77	syl2anc 403	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐 ∈ 𝐴 𝑏 < 𝑐)) ∧ 𝑦 ∈ ℝ) ∧ -𝑎 < 𝑦) ∧ 𝑐 ∈ 𝐴) ∧ -𝑦 < 𝑐) → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴}𝑧 < 𝑦)
79		simpllr 500	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐 ∈ 𝐴 𝑏 < 𝑐)) ∧ 𝑦 ∈ ℝ) → 𝑎 ∈ ℝ)
80		simpr 108	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐 ∈ 𝐴 𝑏 < 𝑐)) ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
81		simplr 496	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐 ∈ 𝐴 𝑏 < 𝑐)) ∧ 𝑦 ∈ ℝ) → ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐 ∈ 𝐴 𝑏 < 𝑐))
82	79, 80, 81	jca31 302	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐 ∈ 𝐴 𝑏 < 𝑐)) ∧ 𝑦 ∈ ℝ) → ((𝑎 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐 ∈ 𝐴 𝑏 < 𝑐)))
83		ltnegcon1 7567	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (-𝑎 < 𝑦 ↔ -𝑦 < 𝑎))
84	83	adantr 270	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐 ∈ 𝐴 𝑏 < 𝑐)) → (-𝑎 < 𝑦 ↔ -𝑦 < 𝑎))
85		renegcl 7369	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℝ → -𝑦 ∈ ℝ)
86		breq1 3788	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = -𝑦 → (𝑏 < 𝑎 ↔ -𝑦 < 𝑎))
87		breq1 3788	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = -𝑦 → (𝑏 < 𝑐 ↔ -𝑦 < 𝑐))
88	87	rexbidv 2369	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = -𝑦 → (∃𝑐 ∈ 𝐴 𝑏 < 𝑐 ↔ ∃𝑐 ∈ 𝐴 -𝑦 < 𝑐))
89	86, 88	imbi12d 232	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = -𝑦 → ((𝑏 < 𝑎 → ∃𝑐 ∈ 𝐴 𝑏 < 𝑐) ↔ (-𝑦 < 𝑎 → ∃𝑐 ∈ 𝐴 -𝑦 < 𝑐)))
90	89	rspcv 2697	. . . . . . . . . . . . . . 15 ⊢ (-𝑦 ∈ ℝ → (∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐 ∈ 𝐴 𝑏 < 𝑐) → (-𝑦 < 𝑎 → ∃𝑐 ∈ 𝐴 -𝑦 < 𝑐)))
91	85, 90	syl 14	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ → (∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐 ∈ 𝐴 𝑏 < 𝑐) → (-𝑦 < 𝑎 → ∃𝑐 ∈ 𝐴 -𝑦 < 𝑐)))
92	91	adantl 271	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐 ∈ 𝐴 𝑏 < 𝑐) → (-𝑦 < 𝑎 → ∃𝑐 ∈ 𝐴 -𝑦 < 𝑐)))
93	92	imp 122	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐 ∈ 𝐴 𝑏 < 𝑐)) → (-𝑦 < 𝑎 → ∃𝑐 ∈ 𝐴 -𝑦 < 𝑐))
94	84, 93	sylbid 148	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐 ∈ 𝐴 𝑏 < 𝑐)) → (-𝑎 < 𝑦 → ∃𝑐 ∈ 𝐴 -𝑦 < 𝑐))
95	94	imp 122	. . . . . . . . . 10 ⊢ ((((𝑎 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐 ∈ 𝐴 𝑏 < 𝑐)) ∧ -𝑎 < 𝑦) → ∃𝑐 ∈ 𝐴 -𝑦 < 𝑐)
96	82, 95	sylan 277	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐 ∈ 𝐴 𝑏 < 𝑐)) ∧ 𝑦 ∈ ℝ) ∧ -𝑎 < 𝑦) → ∃𝑐 ∈ 𝐴 -𝑦 < 𝑐)
97	59, 78, 96	r19.29af 2497	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐 ∈ 𝐴 𝑏 < 𝑐)) ∧ 𝑦 ∈ ℝ) ∧ -𝑎 < 𝑦) → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴}𝑧 < 𝑦)
98	97	ex 113	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐 ∈ 𝐴 𝑏 < 𝑐)) ∧ 𝑦 ∈ ℝ) → (-𝑎 < 𝑦 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴}𝑧 < 𝑦))
99	98	ralrimiva 2434	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐 ∈ 𝐴 𝑏 < 𝑐)) → ∀𝑦 ∈ ℝ (-𝑎 < 𝑦 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴}𝑧 < 𝑦))
100	99	adantrl 461	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ (∀𝑏 ∈ 𝐴 ¬ 𝑎 < 𝑏 ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐 ∈ 𝐴 𝑏 < 𝑐))) → ∀𝑦 ∈ ℝ (-𝑎 < 𝑦 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴}𝑧 < 𝑦))
101		breq2 3789	. . . . . . . . 9 ⊢ (𝑥 = -𝑎 → (𝑦 < 𝑥 ↔ 𝑦 < -𝑎))
102	101	notbid 624	. . . . . . . 8 ⊢ (𝑥 = -𝑎 → (¬ 𝑦 < 𝑥 ↔ ¬ 𝑦 < -𝑎))
103	102	ralbidv 2368	. . . . . . 7 ⊢ (𝑥 = -𝑎 → (∀𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} ¬ 𝑦 < 𝑥 ↔ ∀𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} ¬ 𝑦 < -𝑎))
104		breq1 3788	. . . . . . . . 9 ⊢ (𝑥 = -𝑎 → (𝑥 < 𝑦 ↔ -𝑎 < 𝑦))
105	104	imbi1d 229	. . . . . . . 8 ⊢ (𝑥 = -𝑎 → ((𝑥 < 𝑦 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴}𝑧 < 𝑦) ↔ (-𝑎 < 𝑦 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴}𝑧 < 𝑦)))
106	105	ralbidv 2368	. . . . . . 7 ⊢ (𝑥 = -𝑎 → (∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴}𝑧 < 𝑦) ↔ ∀𝑦 ∈ ℝ (-𝑎 < 𝑦 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴}𝑧 < 𝑦)))
107	103, 106	anbi12d 456	. . . . . 6 ⊢ (𝑥 = -𝑎 → ((∀𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴}𝑧 < 𝑦)) ↔ (∀𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} ¬ 𝑦 < -𝑎 ∧ ∀𝑦 ∈ ℝ (-𝑎 < 𝑦 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴}𝑧 < 𝑦))))
108	107	rspcev 2701	. . . . 5 ⊢ ((-𝑎 ∈ ℝ ∧ (∀𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} ¬ 𝑦 < -𝑎 ∧ ∀𝑦 ∈ ℝ (-𝑎 < 𝑦 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴}𝑧 < 𝑦))) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴}𝑧 < 𝑦)))
109	28, 48, 100, 108	syl12anc 1167	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ (∀𝑏 ∈ 𝐴 ¬ 𝑎 < 𝑏 ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐 ∈ 𝐴 𝑏 < 𝑐))) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴}𝑧 < 𝑦)))
110	109	ex 113	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → ((∀𝑏 ∈ 𝐴 ¬ 𝑎 < 𝑏 ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐 ∈ 𝐴 𝑏 < 𝑐)) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴}𝑧 < 𝑦))))
111	110	rexlimdva 2477	. 2 ⊢ (𝜑 → (∃𝑎 ∈ ℝ (∀𝑏 ∈ 𝐴 ¬ 𝑎 < 𝑏 ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐 ∈ 𝐴 𝑏 < 𝑐)) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴}𝑧 < 𝑦))))
112	26, 111	mpd 13	1 ⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴}𝑧 < 𝑦)))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 102 ↔ wb 103 = wceq 1284 ∈ wcel 1433 ∀wral 2348 ∃wrex 2349 {crab 2352 ⊆ wss 2973 class class class wbr 3785 ℝcr 6980 < clt 7153 -cneg 7280

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-addcom 7076 ax-addass 7078 ax-distr 7080 ax-i2m1 7081 ax-0id 7084 ax-rnegex 7085 ax-cnre 7087 ax-pre-ltadd 7092

This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-pnf 7155 df-mnf 7156 df-ltxr 7158 df-sub 7281 df-neg 7282

This theorem is referenced by: supminfex 8685 infssuzex 10345

W3C validator