Theorem ltsosr 10708

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 10670	. . 3 ⊢ R = ((P × P) / ~R )
2		breq1 5056	. . . 4 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝑓 → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ↔ 𝑓 <R [⟨𝑧, 𝑤⟩] ~R ))
3		eqeq1 2741	. . . . . 6 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝑓 → ([⟨𝑥, 𝑦⟩] ~R = [⟨𝑧, 𝑤⟩] ~R ↔ 𝑓 = [⟨𝑧, 𝑤⟩] ~R ))
4		breq2 5057	. . . . . 6 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝑓 → ([⟨𝑧, 𝑤⟩] ~R <R [⟨𝑥, 𝑦⟩] ~R ↔ [⟨𝑧, 𝑤⟩] ~R <R 𝑓))
5	3, 4	orbi12d 919	. . . . 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 321	. . . 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 349	. . 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 5057	. . . 4 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝑔 → (𝑓 <R [⟨𝑧, 𝑤⟩] ~R ↔ 𝑓 <R 𝑔))
9		eqeq2 2749	. . . . . 6 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝑔 → (𝑓 = [⟨𝑧, 𝑤⟩] ~R ↔ 𝑓 = 𝑔))
10		breq1 5056	. . . . . 6 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝑔 → ([⟨𝑧, 𝑤⟩] ~R <R 𝑓 ↔ 𝑔 <R 𝑓))
11	9, 10	orbi12d 919	. . . . 5 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝑔 → ((𝑓 = [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R 𝑓) ↔ (𝑓 = 𝑔 ∨ 𝑔 <R 𝑓)))
12	11	notbid 321	. . . 4 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝑔 → (¬ (𝑓 = [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R 𝑓) ↔ ¬ (𝑓 = 𝑔 ∨ 𝑔 <R 𝑓)))
13	8, 12	bibi12d 349	. . 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 10691	. . . 4 ⊢ ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ↔ (𝑥 +P 𝑤)<P (𝑦 +P 𝑧))
15		addclpr 10632	. . . . . . 7 ⊢ ((𝑥 ∈ P ∧ 𝑤 ∈ P) → (𝑥 +P 𝑤) ∈ P)
16		addclpr 10632	. . . . . . 7 ⊢ ((𝑦 ∈ P ∧ 𝑧 ∈ P) → (𝑦 +P 𝑧) ∈ P)
17		ltsopr 10646	. . . . . . . 8 ⊢ <P Or P
18		sotric 5496	. . . . . . . 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 690	. . . . . . 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 599	. . . . . 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 661	. . . . 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 10680	. . . . . . 7 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ([⟨𝑥, 𝑦⟩] ~R = [⟨𝑧, 𝑤⟩] ~R ↔ (𝑥 +P 𝑤) = (𝑦 +P 𝑧)))
23		ltsrpr 10691	. . . . . . . . 9 ⊢ ([⟨𝑧, 𝑤⟩] ~R <R [⟨𝑥, 𝑦⟩] ~R ↔ (𝑧 +P 𝑦)<P (𝑤 +P 𝑥))
24		addcompr 10635	. . . . . . . . . 10 ⊢ (𝑧 +P 𝑦) = (𝑦 +P 𝑧)
25		addcompr 10635	. . . . . . . . . 10 ⊢ (𝑤 +P 𝑥) = (𝑥 +P 𝑤)
26	24, 25	breq12i 5062	. . . . . . . . 9 ⊢ ((𝑧 +P 𝑦)<P (𝑤 +P 𝑥) ↔ (𝑦 +P 𝑧)<P (𝑥 +P 𝑤))
27	23, 26	bitri 278	. . . . . . . 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 919	. . . . . 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 321	. . . . 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 285	. . . 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	syl5bb 286	. . 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 8490	. 2 ⊢ ((𝑓 ∈ R ∧ 𝑔 ∈ R) → (𝑓 <R 𝑔 ↔ ¬ (𝑓 = 𝑔 ∨ 𝑔 <R 𝑓)))
34	2	anbi1d 633	. . . 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 5056	. . . 4 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝑓 → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ↔ 𝑓 <R [⟨𝑣, 𝑢⟩] ~R ))
36	34, 35	imbi12d 348	. . 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 5056	. . . . 5 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝑔 → ([⟨𝑧, 𝑤⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ↔ 𝑔 <R [⟨𝑣, 𝑢⟩] ~R ))
38	8, 37	anbi12d 634	. . . 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 345	. . 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 5057	. . . . 5 ⊢ ([⟨𝑣, 𝑢⟩] ~R = ℎ → (𝑔 <R [⟨𝑣, 𝑢⟩] ~R ↔ 𝑔 <R ℎ))
41	40	anbi2d 632	. . . 4 ⊢ ([⟨𝑣, 𝑢⟩] ~R = ℎ → ((𝑓 <R 𝑔 ∧ 𝑔 <R [⟨𝑣, 𝑢⟩] ~R ) ↔ (𝑓 <R 𝑔 ∧ 𝑔 <R ℎ)))
42		breq2 5057	. . . 4 ⊢ ([⟨𝑣, 𝑢⟩] ~R = ℎ → (𝑓 <R [⟨𝑣, 𝑢⟩] ~R ↔ 𝑓 <R ℎ))
43	41, 42	imbi12d 348	. . 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 7246	. . . . . . . . . 10 ⊢ (𝑥 +P 𝑤) ∈ V
45		ovex 7246	. . . . . . . . . 10 ⊢ (𝑦 +P 𝑧) ∈ V
46		ltapr 10659	. . . . . . . . . 10 ⊢ (ℎ ∈ P → (𝑓<P 𝑔 ↔ (ℎ +P 𝑓)<P (ℎ +P 𝑔)))
47		vex 3412	. . . . . . . . . 10 ⊢ 𝑢 ∈ V
48		addcompr 10635	. . . . . . . . . 10 ⊢ (𝑓 +P 𝑔) = (𝑔 +P 𝑓)
49	44, 45, 46, 47, 48	caovord2 7420	. . . . . . . . 9 ⊢ (𝑢 ∈ P → ((𝑥 +P 𝑤)<P (𝑦 +P 𝑧) ↔ ((𝑥 +P 𝑤) +P 𝑢)<P ((𝑦 +P 𝑧) +P 𝑢)))
50		addasspr 10636	. . . . . . . . . 10 ⊢ ((𝑥 +P 𝑤) +P 𝑢) = (𝑥 +P (𝑤 +P 𝑢))
51		addasspr 10636	. . . . . . . . . 10 ⊢ ((𝑦 +P 𝑧) +P 𝑢) = (𝑦 +P (𝑧 +P 𝑢))
52	50, 51	breq12i 5062	. . . . . . . . 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 290	. . . . . . . 8 ⊢ (𝑢 ∈ P → ((𝑥 +P 𝑤)<P (𝑦 +P 𝑧) ↔ (𝑥 +P (𝑤 +P 𝑢))<P (𝑦 +P (𝑧 +P 𝑢))))
54	14, 53	syl5bb 286	. . . . . . 7 ⊢ (𝑢 ∈ P → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ↔ (𝑥 +P (𝑤 +P 𝑢))<P (𝑦 +P (𝑧 +P 𝑢))))
55		ltsrpr 10691	. . . . . . . 8 ⊢ ([⟨𝑧, 𝑤⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ↔ (𝑧 +P 𝑢)<P (𝑤 +P 𝑣))
56		ltapr 10659	. . . . . . . 8 ⊢ (𝑦 ∈ P → ((𝑧 +P 𝑢)<P (𝑤 +P 𝑣) ↔ (𝑦 +P (𝑧 +P 𝑢))<P (𝑦 +P (𝑤 +P 𝑣))))
57	55, 56	syl5bb 286	. . . . . . 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 640	. . . . . 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 10612	. . . . . . . 8 ⊢ <P ⊆ (P × P)
60	17, 59	sotri 5992	. . . . . . 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 10626	. . . . . . . . 9 ⊢ dom +P = (P × P)
62		0npr 10606	. . . . . . . . 9 ⊢ ¬ ∅ ∈ P
63		ltapr 10659	. . . . . . . . 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 7399	. . . . . . . 8 ⊢ ((𝑤 +P (𝑥 +P 𝑢))<P (𝑤 +P (𝑦 +P 𝑣)) → (𝑥 +P 𝑢)<P (𝑦 +P 𝑣))
65		vex 3412	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
66		vex 3412	. . . . . . . . . 10 ⊢ 𝑤 ∈ V
67		addasspr 10636	. . . . . . . . . 10 ⊢ ((𝑓 +P 𝑔) +P ℎ) = (𝑓 +P (𝑔 +P ℎ))
68	65, 66, 47, 48, 67	caov12 7436	. . . . . . . . 9 ⊢ (𝑥 +P (𝑤 +P 𝑢)) = (𝑤 +P (𝑥 +P 𝑢))
69		vex 3412	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
70		vex 3412	. . . . . . . . . 10 ⊢ 𝑣 ∈ V
71	69, 66, 70, 48, 67	caov12 7436	. . . . . . . . 9 ⊢ (𝑦 +P (𝑤 +P 𝑣)) = (𝑤 +P (𝑦 +P 𝑣))
72	68, 71	breq12i 5062	. . . . . . . 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 10691	. . . . . . . 8 ⊢ ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ↔ (𝑥 +P 𝑢)<P (𝑦 +P 𝑣))
74	64, 72, 73	3imtr4i 295	. . . . . . 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	syl6bi 256	. . . . 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 746	. . . 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 1133	. . 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 8491	. 2 ⊢ ((𝑓 ∈ R ∧ 𝑔 ∈ R ∧ ℎ ∈ R) → ((𝑓 <R 𝑔 ∧ 𝑔 <R ℎ) → 𝑓 <R ℎ))
80	33, 79	isso2i 5503	1 ⊢ <R Or R

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 847   = wceq 1543   ∈ wcel 2110  ⟨cop 4547   class class class wbr 5053   Or wor 5467  (class class class)co 7213  [cec 8389  Pcnp 10473   +_P cpp 10475  <_P cltp 10477   ~_R cer 10478  Rcnr 10479   <_R cltr 10485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-inf2 9256

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-oadd 8206  df-omul 8207  df-er 8391  df-ec 8393  df-qs 8397  df-ni 10486  df-pli 10487  df-mi 10488  df-lti 10489  df-plpq 10522  df-mpq 10523  df-ltpq 10524  df-enq 10525  df-nq 10526  df-erq 10527  df-plq 10528  df-mq 10529  df-1nq 10530  df-rq 10531  df-ltnq 10532  df-np 10595  df-plp 10597  df-ltp 10599  df-enr 10669  df-nr 10670  df-ltr 10673

This theorem is referenced by:  1ne0sr  10710  addgt0sr  10718  sqgt0sr  10720  supsrlem  10725  axpre-lttri  10779  axpre-lttrn  10780