Theorem ltsosr 11117

Description: Signed real 'less than' is a strict ordering. (Contributed by NM, 19-Feb-1996.) (New usage is discouraged.)

Assertion

Ref	Expression
ltsosr	⊢ <R Or R

Proof of Theorem ltsosr

Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑣 𝑢 𝑓 𝑔 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-nr 11079	. . 3 ⊢ R = ((P × P) / ~R )
2		breq1 5151	. . . 4 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝑓 → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ↔ 𝑓 <R [⟨𝑧, 𝑤⟩] ~R ))
3		eqeq1 2729	. . . . . 6 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝑓 → ([⟨𝑥, 𝑦⟩] ~R = [⟨𝑧, 𝑤⟩] ~R ↔ 𝑓 = [⟨𝑧, 𝑤⟩] ~R ))
4		breq2 5152	. . . . . 6 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝑓 → ([⟨𝑧, 𝑤⟩] ~R <R [⟨𝑥, 𝑦⟩] ~R ↔ [⟨𝑧, 𝑤⟩] ~R <R 𝑓))
5	3, 4	orbi12d 916	. . . . 5 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝑓 → (([⟨𝑥, 𝑦⟩] ~R = [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R [⟨𝑥, 𝑦⟩] ~R ) ↔ (𝑓 = [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R 𝑓)))
6	5	notbid 317	. . . 4 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝑓 → (¬ ([⟨𝑥, 𝑦⟩] ~R = [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R [⟨𝑥, 𝑦⟩] ~R ) ↔ ¬ (𝑓 = [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R 𝑓)))
7	2, 6	bibi12d 344	. . 3 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝑓 → (([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ↔ ¬ ([⟨𝑥, 𝑦⟩] ~R = [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R [⟨𝑥, 𝑦⟩] ~R )) ↔ (𝑓 <R [⟨𝑧, 𝑤⟩] ~R ↔ ¬ (𝑓 = [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R 𝑓))))
8		breq2 5152	. . . 4 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝑔 → (𝑓 <R [⟨𝑧, 𝑤⟩] ~R ↔ 𝑓 <R 𝑔))
9		eqeq2 2737	. . . . . 6 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝑔 → (𝑓 = [⟨𝑧, 𝑤⟩] ~R ↔ 𝑓 = 𝑔))
10		breq1 5151	. . . . . 6 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝑔 → ([⟨𝑧, 𝑤⟩] ~R <R 𝑓 ↔ 𝑔 <R 𝑓))
11	9, 10	orbi12d 916	. . . . 5 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝑔 → ((𝑓 = [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R 𝑓) ↔ (𝑓 = 𝑔 ∨ 𝑔 <R 𝑓)))
12	11	notbid 317	. . . 4 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝑔 → (¬ (𝑓 = [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R 𝑓) ↔ ¬ (𝑓 = 𝑔 ∨ 𝑔 <R 𝑓)))
13	8, 12	bibi12d 344	. . 3 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝑔 → ((𝑓 <R [⟨𝑧, 𝑤⟩] ~R ↔ ¬ (𝑓 = [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R 𝑓)) ↔ (𝑓 <R 𝑔 ↔ ¬ (𝑓 = 𝑔 ∨ 𝑔 <R 𝑓))))
14		ltsrpr 11100	. . . 4 ⊢ ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ↔ (𝑥 +P 𝑤)<P (𝑦 +P 𝑧))
15		addclpr 11041	. . . . . . 7 ⊢ ((𝑥 ∈ P ∧ 𝑤 ∈ P) → (𝑥 +P 𝑤) ∈ P)
16		addclpr 11041	. . . . . . 7 ⊢ ((𝑦 ∈ P ∧ 𝑧 ∈ P) → (𝑦 +P 𝑧) ∈ P)
17		ltsopr 11055	. . . . . . . 8 ⊢ <P Or P
18		sotric 5617	. . . . . . . 8 ⊢ ((<P Or P ∧ ((𝑥 +P 𝑤) ∈ P ∧ (𝑦 +P 𝑧) ∈ P)) → ((𝑥 +P 𝑤)<P (𝑦 +P 𝑧) ↔ ¬ ((𝑥 +P 𝑤) = (𝑦 +P 𝑧) ∨ (𝑦 +P 𝑧)<P (𝑥 +P 𝑤))))
19	17, 18	mpan 688	. . . . . . 7 ⊢ (((𝑥 +P 𝑤) ∈ P ∧ (𝑦 +P 𝑧) ∈ P) → ((𝑥 +P 𝑤)<P (𝑦 +P 𝑧) ↔ ¬ ((𝑥 +P 𝑤) = (𝑦 +P 𝑧) ∨ (𝑦 +P 𝑧)<P (𝑥 +P 𝑤))))
20	15, 16, 19	syl2an 594	. . . . . 6 ⊢ (((𝑥 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑦 ∈ P ∧ 𝑧 ∈ P)) → ((𝑥 +P 𝑤)<P (𝑦 +P 𝑧) ↔ ¬ ((𝑥 +P 𝑤) = (𝑦 +P 𝑧) ∨ (𝑦 +P 𝑧)<P (𝑥 +P 𝑤))))
21	20	an42s 659	. . . . 5 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ((𝑥 +P 𝑤)<P (𝑦 +P 𝑧) ↔ ¬ ((𝑥 +P 𝑤) = (𝑦 +P 𝑧) ∨ (𝑦 +P 𝑧)<P (𝑥 +P 𝑤))))
22		enreceq 11089	. . . . . . 7 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ([⟨𝑥, 𝑦⟩] ~R = [⟨𝑧, 𝑤⟩] ~R ↔ (𝑥 +P 𝑤) = (𝑦 +P 𝑧)))
23		ltsrpr 11100	. . . . . . . . 9 ⊢ ([⟨𝑧, 𝑤⟩] ~R <R [⟨𝑥, 𝑦⟩] ~R ↔ (𝑧 +P 𝑦)<P (𝑤 +P 𝑥))
24		addcompr 11044	. . . . . . . . . 10 ⊢ (𝑧 +P 𝑦) = (𝑦 +P 𝑧)
25		addcompr 11044	. . . . . . . . . 10 ⊢ (𝑤 +P 𝑥) = (𝑥 +P 𝑤)
26	24, 25	breq12i 5157	. . . . . . . . 9 ⊢ ((𝑧 +P 𝑦)<P (𝑤 +P 𝑥) ↔ (𝑦 +P 𝑧)<P (𝑥 +P 𝑤))
27	23, 26	bitri 274	. . . . . . . 8 ⊢ ([⟨𝑧, 𝑤⟩] ~R <R [⟨𝑥, 𝑦⟩] ~R ↔ (𝑦 +P 𝑧)<P (𝑥 +P 𝑤))
28	27	a1i 11	. . . . . . 7 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ([⟨𝑧, 𝑤⟩] ~R <R [⟨𝑥, 𝑦⟩] ~R ↔ (𝑦 +P 𝑧)<P (𝑥 +P 𝑤)))
29	22, 28	orbi12d 916	. . . . . 6 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → (([⟨𝑥, 𝑦⟩] ~R = [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R [⟨𝑥, 𝑦⟩] ~R ) ↔ ((𝑥 +P 𝑤) = (𝑦 +P 𝑧) ∨ (𝑦 +P 𝑧)<P (𝑥 +P 𝑤))))
30	29	notbid 317	. . . . 5 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → (¬ ([⟨𝑥, 𝑦⟩] ~R = [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R [⟨𝑥, 𝑦⟩] ~R ) ↔ ¬ ((𝑥 +P 𝑤) = (𝑦 +P 𝑧) ∨ (𝑦 +P 𝑧)<P (𝑥 +P 𝑤))))
31	21, 30	bitr4d 281	. . . 4 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ((𝑥 +P 𝑤)<P (𝑦 +P 𝑧) ↔ ¬ ([⟨𝑥, 𝑦⟩] ~R = [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R [⟨𝑥, 𝑦⟩] ~R )))
32	14, 31	bitrid 282	. . 3 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ↔ ¬ ([⟨𝑥, 𝑦⟩] ~R = [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R [⟨𝑥, 𝑦⟩] ~R )))
33	1, 7, 13, 32	2ecoptocl 8825	. 2 ⊢ ((𝑓 ∈ R ∧ 𝑔 ∈ R) → (𝑓 <R 𝑔 ↔ ¬ (𝑓 = 𝑔 ∨ 𝑔 <R 𝑓)))
34	2	anbi1d 629	. . . 4 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝑓 → (([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ∧ [⟨𝑧, 𝑤⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ) ↔ (𝑓 <R [⟨𝑧, 𝑤⟩] ~R ∧ [⟨𝑧, 𝑤⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R )))
35		breq1 5151	. . . 4 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝑓 → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ↔ 𝑓 <R [⟨𝑣, 𝑢⟩] ~R ))
36	34, 35	imbi12d 343	. . 3 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝑓 → ((([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ∧ [⟨𝑧, 𝑤⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ) → [⟨𝑥, 𝑦⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ) ↔ ((𝑓 <R [⟨𝑧, 𝑤⟩] ~R ∧ [⟨𝑧, 𝑤⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ) → 𝑓 <R [⟨𝑣, 𝑢⟩] ~R )))
37		breq1 5151	. . . . 5 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝑔 → ([⟨𝑧, 𝑤⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ↔ 𝑔 <R [⟨𝑣, 𝑢⟩] ~R ))
38	8, 37	anbi12d 630	. . . 4 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝑔 → ((𝑓 <R [⟨𝑧, 𝑤⟩] ~R ∧ [⟨𝑧, 𝑤⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ) ↔ (𝑓 <R 𝑔 ∧ 𝑔 <R [⟨𝑣, 𝑢⟩] ~R )))
39	38	imbi1d 340	. . 3 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝑔 → (((𝑓 <R [⟨𝑧, 𝑤⟩] ~R ∧ [⟨𝑧, 𝑤⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ) → 𝑓 <R [⟨𝑣, 𝑢⟩] ~R ) ↔ ((𝑓 <R 𝑔 ∧ 𝑔 <R [⟨𝑣, 𝑢⟩] ~R ) → 𝑓 <R [⟨𝑣, 𝑢⟩] ~R )))
40		breq2 5152	. . . . 5 ⊢ ([⟨𝑣, 𝑢⟩] ~R = ℎ → (𝑔 <R [⟨𝑣, 𝑢⟩] ~R ↔ 𝑔 <R ℎ))
41	40	anbi2d 628	. . . 4 ⊢ ([⟨𝑣, 𝑢⟩] ~R = ℎ → ((𝑓 <R 𝑔 ∧ 𝑔 <R [⟨𝑣, 𝑢⟩] ~R ) ↔ (𝑓 <R 𝑔 ∧ 𝑔 <R ℎ)))
42		breq2 5152	. . . 4 ⊢ ([⟨𝑣, 𝑢⟩] ~R = ℎ → (𝑓 <R [⟨𝑣, 𝑢⟩] ~R ↔ 𝑓 <R ℎ))
43	41, 42	imbi12d 343	. . 3 ⊢ ([⟨𝑣, 𝑢⟩] ~R = ℎ → (((𝑓 <R 𝑔 ∧ 𝑔 <R [⟨𝑣, 𝑢⟩] ~R ) → 𝑓 <R [⟨𝑣, 𝑢⟩] ~R ) ↔ ((𝑓 <R 𝑔 ∧ 𝑔 <R ℎ) → 𝑓 <R ℎ)))
44		ovex 7450	. . . . . . . . . 10 ⊢ (𝑥 +P 𝑤) ∈ V
45		ovex 7450	. . . . . . . . . 10 ⊢ (𝑦 +P 𝑧) ∈ V
46		ltapr 11068	. . . . . . . . . 10 ⊢ (ℎ ∈ P → (𝑓<P 𝑔 ↔ (ℎ +P 𝑓)<P (ℎ +P 𝑔)))
47		vex 3467	. . . . . . . . . 10 ⊢ 𝑢 ∈ V
48		addcompr 11044	. . . . . . . . . 10 ⊢ (𝑓 +P 𝑔) = (𝑔 +P 𝑓)
49	44, 45, 46, 47, 48	caovord2 7631	. . . . . . . . 9 ⊢ (𝑢 ∈ P → ((𝑥 +P 𝑤)<P (𝑦 +P 𝑧) ↔ ((𝑥 +P 𝑤) +P 𝑢)<P ((𝑦 +P 𝑧) +P 𝑢)))
50		addasspr 11045	. . . . . . . . . 10 ⊢ ((𝑥 +P 𝑤) +P 𝑢) = (𝑥 +P (𝑤 +P 𝑢))
51		addasspr 11045	. . . . . . . . . 10 ⊢ ((𝑦 +P 𝑧) +P 𝑢) = (𝑦 +P (𝑧 +P 𝑢))
52	50, 51	breq12i 5157	. . . . . . . . 9 ⊢ (((𝑥 +P 𝑤) +P 𝑢)<P ((𝑦 +P 𝑧) +P 𝑢) ↔ (𝑥 +P (𝑤 +P 𝑢))<P (𝑦 +P (𝑧 +P 𝑢)))
53	49, 52	bitrdi 286	. . . . . . . 8 ⊢ (𝑢 ∈ P → ((𝑥 +P 𝑤)<P (𝑦 +P 𝑧) ↔ (𝑥 +P (𝑤 +P 𝑢))<P (𝑦 +P (𝑧 +P 𝑢))))
54	14, 53	bitrid 282	. . . . . . 7 ⊢ (𝑢 ∈ P → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ↔ (𝑥 +P (𝑤 +P 𝑢))<P (𝑦 +P (𝑧 +P 𝑢))))
55		ltsrpr 11100	. . . . . . . 8 ⊢ ([⟨𝑧, 𝑤⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ↔ (𝑧 +P 𝑢)<P (𝑤 +P 𝑣))
56		ltapr 11068	. . . . . . . 8 ⊢ (𝑦 ∈ P → ((𝑧 +P 𝑢)<P (𝑤 +P 𝑣) ↔ (𝑦 +P (𝑧 +P 𝑢))<P (𝑦 +P (𝑤 +P 𝑣))))
57	55, 56	bitrid 282	. . . . . . 7 ⊢ (𝑦 ∈ P → ([⟨𝑧, 𝑤⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ↔ (𝑦 +P (𝑧 +P 𝑢))<P (𝑦 +P (𝑤 +P 𝑣))))
58	54, 57	bi2anan9r 637	. . . . . 6 ⊢ ((𝑦 ∈ P ∧ 𝑢 ∈ P) → (([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ∧ [⟨𝑧, 𝑤⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ) ↔ ((𝑥 +P (𝑤 +P 𝑢))<P (𝑦 +P (𝑧 +P 𝑢)) ∧ (𝑦 +P (𝑧 +P 𝑢))<P (𝑦 +P (𝑤 +P 𝑣)))))
59		ltrelpr 11021	. . . . . . . 8 ⊢ <P ⊆ (P × P)
60	17, 59	sotri 6133	. . . . . . 7 ⊢ (((𝑥 +P (𝑤 +P 𝑢))<P (𝑦 +P (𝑧 +P 𝑢)) ∧ (𝑦 +P (𝑧 +P 𝑢))<P (𝑦 +P (𝑤 +P 𝑣))) → (𝑥 +P (𝑤 +P 𝑢))<P (𝑦 +P (𝑤 +P 𝑣)))
61		dmplp 11035	. . . . . . . . 9 ⊢ dom +P = (P × P)
62		0npr 11015	. . . . . . . . 9 ⊢ ¬ ∅ ∈ P
63		ltapr 11068	. . . . . . . . 9 ⊢ (𝑤 ∈ P → ((𝑥 +P 𝑢)<P (𝑦 +P 𝑣) ↔ (𝑤 +P (𝑥 +P 𝑢))<P (𝑤 +P (𝑦 +P 𝑣))))
64	61, 59, 62, 63	ndmovordi 7610	. . . . . . . 8 ⊢ ((𝑤 +P (𝑥 +P 𝑢))<P (𝑤 +P (𝑦 +P 𝑣)) → (𝑥 +P 𝑢)<P (𝑦 +P 𝑣))
65		vex 3467	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
66		vex 3467	. . . . . . . . . 10 ⊢ 𝑤 ∈ V
67		addasspr 11045	. . . . . . . . . 10 ⊢ ((𝑓 +P 𝑔) +P ℎ) = (𝑓 +P (𝑔 +P ℎ))
68	65, 66, 47, 48, 67	caov12 7647	. . . . . . . . 9 ⊢ (𝑥 +P (𝑤 +P 𝑢)) = (𝑤 +P (𝑥 +P 𝑢))
69		vex 3467	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
70		vex 3467	. . . . . . . . . 10 ⊢ 𝑣 ∈ V
71	69, 66, 70, 48, 67	caov12 7647	. . . . . . . . 9 ⊢ (𝑦 +P (𝑤 +P 𝑣)) = (𝑤 +P (𝑦 +P 𝑣))
72	68, 71	breq12i 5157	. . . . . . . 8 ⊢ ((𝑥 +P (𝑤 +P 𝑢))<P (𝑦 +P (𝑤 +P 𝑣)) ↔ (𝑤 +P (𝑥 +P 𝑢))<P (𝑤 +P (𝑦 +P 𝑣)))
73		ltsrpr 11100	. . . . . . . 8 ⊢ ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ↔ (𝑥 +P 𝑢)<P (𝑦 +P 𝑣))
74	64, 72, 73	3imtr4i 291	. . . . . . 7 ⊢ ((𝑥 +P (𝑤 +P 𝑢))<P (𝑦 +P (𝑤 +P 𝑣)) → [⟨𝑥, 𝑦⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R )
75	60, 74	syl 17	. . . . . 6 ⊢ (((𝑥 +P (𝑤 +P 𝑢))<P (𝑦 +P (𝑧 +P 𝑢)) ∧ (𝑦 +P (𝑧 +P 𝑢))<P (𝑦 +P (𝑤 +P 𝑣))) → [⟨𝑥, 𝑦⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R )
76	58, 75	biimtrdi 252	. . . . 5 ⊢ ((𝑦 ∈ P ∧ 𝑢 ∈ P) → (([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ∧ [⟨𝑧, 𝑤⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ) → [⟨𝑥, 𝑦⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ))
77	76	ad2ant2l 744	. . . 4 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → (([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ∧ [⟨𝑧, 𝑤⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ) → [⟨𝑥, 𝑦⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ))
78	77	3adant2 1128	. . 3 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → (([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ∧ [⟨𝑧, 𝑤⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ) → [⟨𝑥, 𝑦⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ))
79	1, 36, 39, 43, 78	3ecoptocl 8826	. 2 ⊢ ((𝑓 ∈ R ∧ 𝑔 ∈ R ∧ ℎ ∈ R) → ((𝑓 <R 𝑔 ∧ 𝑔 <R ℎ) → 𝑓 <R ℎ))
80	33, 79	isso2i 5624	1 ⊢ <R Or R

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394   ∨ wo 845   = wceq 1533   ∈ wcel 2098  ⟨cop 4635   class class class wbr 5148   Or wor 5588  (class class class)co 7417  [cec 8721  Pcnp 10882   +_P cpp 10884  <_P cltp 10886   ~_R cer 10887  Rcnr 10888   <_R cltr 10894

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5299  ax-nul 5306  ax-pow 5364  ax-pr 5428  ax-un 7739  ax-inf2 9664

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3965  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6499  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-ov 7420  df-oprab 7421  df-mpo 7422  df-om 7870  df-1st 7992  df-2nd 7993  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-oadd 8489  df-omul 8490  df-er 8723  df-ec 8725  df-qs 8729  df-ni 10895  df-pli 10896  df-mi 10897  df-lti 10898  df-plpq 10931  df-mpq 10932  df-ltpq 10933  df-enq 10934  df-nq 10935  df-erq 10936  df-plq 10937  df-mq 10938  df-1nq 10939  df-rq 10940  df-ltnq 10941  df-np 11004  df-plp 11006  df-ltp 11008  df-enr 11078  df-nr 11079  df-ltr 11082

This theorem is referenced by:  1ne0sr  11119  addgt0sr  11127  sqgt0sr  11129  supsrlem  11134  axpre-lttri  11188  axpre-lttrn  11189