Theorem ltsopr 7656

Description: Positive real 'less than' is a weak linear order (in the sense of df-iso 4328). Proposition 11.2.3 of [HoTT], p. (varies). (Contributed by Jim Kingdon, 16-Dec-2019.)

Assertion

Ref	Expression
ltsopr	⊢ <P Or P

Proof of Theorem ltsopr

Dummy variables 𝑟 𝑞 𝑠 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltpopr 7655	. 2 ⊢ <P Po P
2		ltdfpr 7566	. . . . 5 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → (𝑥<P 𝑦 ↔ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦))))
3	2	3adant3 1019	. . . 4 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P ∧ 𝑧 ∈ P) → (𝑥<P 𝑦 ↔ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦))))
4		prop 7535	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ P → ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ P)
5		prnminu 7549	. . . . . . . . . . . 12 ⊢ ((⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ P ∧ 𝑞 ∈ (2nd ‘𝑥)) → ∃𝑟 ∈ (2nd ‘𝑥)𝑟 <Q 𝑞)
6	4, 5	sylan 283	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ P ∧ 𝑞 ∈ (2nd ‘𝑥)) → ∃𝑟 ∈ (2nd ‘𝑥)𝑟 <Q 𝑞)
7		prop 7535	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ P → ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ ∈ P)
8		prnmaxl 7548	. . . . . . . . . . . 12 ⊢ ((⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ ∈ P ∧ 𝑞 ∈ (1st ‘𝑦)) → ∃𝑠 ∈ (1st ‘𝑦)𝑞 <Q 𝑠)
9	7, 8	sylan 283	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ P ∧ 𝑞 ∈ (1st ‘𝑦)) → ∃𝑠 ∈ (1st ‘𝑦)𝑞 <Q 𝑠)
10	6, 9	anim12i 338	. . . . . . . . . 10 ⊢ (((𝑥 ∈ P ∧ 𝑞 ∈ (2nd ‘𝑥)) ∧ (𝑦 ∈ P ∧ 𝑞 ∈ (1st ‘𝑦))) → (∃𝑟 ∈ (2nd ‘𝑥)𝑟 <Q 𝑞 ∧ ∃𝑠 ∈ (1st ‘𝑦)𝑞 <Q 𝑠))
11	10	an4s 588	. . . . . . . . 9 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦))) → (∃𝑟 ∈ (2nd ‘𝑥)𝑟 <Q 𝑞 ∧ ∃𝑠 ∈ (1st ‘𝑦)𝑞 <Q 𝑠))
12		reeanv 2664	. . . . . . . . 9 ⊢ (∃𝑟 ∈ (2nd ‘𝑥)∃𝑠 ∈ (1st ‘𝑦)(𝑟 <Q 𝑞 ∧ 𝑞 <Q 𝑠) ↔ (∃𝑟 ∈ (2nd ‘𝑥)𝑟 <Q 𝑞 ∧ ∃𝑠 ∈ (1st ‘𝑦)𝑞 <Q 𝑠))
13	11, 12	sylibr 134	. . . . . . . 8 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦))) → ∃𝑟 ∈ (2nd ‘𝑥)∃𝑠 ∈ (1st ‘𝑦)(𝑟 <Q 𝑞 ∧ 𝑞 <Q 𝑠))
14	13	3adantl3 1157	. . . . . . 7 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P ∧ 𝑧 ∈ P) ∧ (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦))) → ∃𝑟 ∈ (2nd ‘𝑥)∃𝑠 ∈ (1st ‘𝑦)(𝑟 <Q 𝑞 ∧ 𝑞 <Q 𝑠))
15		ltsonq 7458	. . . . . . . . . . . . 13 ⊢ <Q Or Q
16		ltrelnq 7425	. . . . . . . . . . . . 13 ⊢ <Q ⊆ (Q × Q)
17	15, 16	sotri 5061	. . . . . . . . . . . 12 ⊢ ((𝑟 <Q 𝑞 ∧ 𝑞 <Q 𝑠) → 𝑟 <Q 𝑠)
18	17	adantl 277	. . . . . . . . . . 11 ⊢ (((((𝑥 ∈ P ∧ 𝑦 ∈ P ∧ 𝑧 ∈ P) ∧ (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦))) ∧ (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑠 ∈ (1st ‘𝑦))) ∧ (𝑟 <Q 𝑞 ∧ 𝑞 <Q 𝑠)) → 𝑟 <Q 𝑠)
19		prop 7535	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ P → ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∈ P)
20		prloc 7551	. . . . . . . . . . . . . . . 16 ⊢ ((⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∈ P ∧ 𝑟 <Q 𝑠) → (𝑟 ∈ (1st ‘𝑧) ∨ 𝑠 ∈ (2nd ‘𝑧)))
21	19, 20	sylan 283	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ P ∧ 𝑟 <Q 𝑠) → (𝑟 ∈ (1st ‘𝑧) ∨ 𝑠 ∈ (2nd ‘𝑧)))
22	21	3ad2antl3 1163	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P ∧ 𝑧 ∈ P) ∧ 𝑟 <Q 𝑠) → (𝑟 ∈ (1st ‘𝑧) ∨ 𝑠 ∈ (2nd ‘𝑧)))
23	22	ex 115	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P ∧ 𝑧 ∈ P) → (𝑟 <Q 𝑠 → (𝑟 ∈ (1st ‘𝑧) ∨ 𝑠 ∈ (2nd ‘𝑧))))
24	23	adantr 276	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P ∧ 𝑧 ∈ P) ∧ (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦))) → (𝑟 <Q 𝑠 → (𝑟 ∈ (1st ‘𝑧) ∨ 𝑠 ∈ (2nd ‘𝑧))))
25	24	ad2antrr 488	. . . . . . . . . . 11 ⊢ (((((𝑥 ∈ P ∧ 𝑦 ∈ P ∧ 𝑧 ∈ P) ∧ (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦))) ∧ (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑠 ∈ (1st ‘𝑦))) ∧ (𝑟 <Q 𝑞 ∧ 𝑞 <Q 𝑠)) → (𝑟 <Q 𝑠 → (𝑟 ∈ (1st ‘𝑧) ∨ 𝑠 ∈ (2nd ‘𝑧))))
26	18, 25	mpd 13	. . . . . . . . . 10 ⊢ (((((𝑥 ∈ P ∧ 𝑦 ∈ P ∧ 𝑧 ∈ P) ∧ (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦))) ∧ (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑠 ∈ (1st ‘𝑦))) ∧ (𝑟 <Q 𝑞 ∧ 𝑞 <Q 𝑠)) → (𝑟 ∈ (1st ‘𝑧) ∨ 𝑠 ∈ (2nd ‘𝑧)))
27		elprnqu 7542	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ P ∧ 𝑟 ∈ (2nd ‘𝑥)) → 𝑟 ∈ Q)
28	4, 27	sylan 283	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ P ∧ 𝑟 ∈ (2nd ‘𝑥)) → 𝑟 ∈ Q)
29		ax-ia3 108	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑟 ∈ (2nd ‘𝑥) → (𝑟 ∈ (1st ‘𝑧) → (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑟 ∈ (1st ‘𝑧))))
30	29	adantl 277	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ P ∧ 𝑟 ∈ (2nd ‘𝑥)) → (𝑟 ∈ (1st ‘𝑧) → (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑟 ∈ (1st ‘𝑧))))
31		19.8a 1601	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑟 ∈ Q ∧ (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑟 ∈ (1st ‘𝑧))) → ∃𝑟(𝑟 ∈ Q ∧ (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑟 ∈ (1st ‘𝑧))))
32	28, 30, 31	syl6an 1445	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ P ∧ 𝑟 ∈ (2nd ‘𝑥)) → (𝑟 ∈ (1st ‘𝑧) → ∃𝑟(𝑟 ∈ Q ∧ (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑟 ∈ (1st ‘𝑧)))))
33	32	3ad2antl1 1161	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P ∧ 𝑧 ∈ P) ∧ 𝑟 ∈ (2nd ‘𝑥)) → (𝑟 ∈ (1st ‘𝑧) → ∃𝑟(𝑟 ∈ Q ∧ (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑟 ∈ (1st ‘𝑧)))))
34	33	imp 124	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑥 ∈ P ∧ 𝑦 ∈ P ∧ 𝑧 ∈ P) ∧ 𝑟 ∈ (2nd ‘𝑥)) ∧ 𝑟 ∈ (1st ‘𝑧)) → ∃𝑟(𝑟 ∈ Q ∧ (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑟 ∈ (1st ‘𝑧))))
35		df-rex 2478	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑟 ∈ (1st ‘𝑧)) ↔ ∃𝑟(𝑟 ∈ Q ∧ (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑟 ∈ (1st ‘𝑧))))
36	34, 35	sylibr 134	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥 ∈ P ∧ 𝑦 ∈ P ∧ 𝑧 ∈ P) ∧ 𝑟 ∈ (2nd ‘𝑥)) ∧ 𝑟 ∈ (1st ‘𝑧)) → ∃𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑟 ∈ (1st ‘𝑧)))
37		ltdfpr 7566	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ P ∧ 𝑧 ∈ P) → (𝑥<P 𝑧 ↔ ∃𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑟 ∈ (1st ‘𝑧))))
38	37	biimprd 158	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ P ∧ 𝑧 ∈ P) → (∃𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑟 ∈ (1st ‘𝑧)) → 𝑥<P 𝑧))
39	38	3adant2 1018	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P ∧ 𝑧 ∈ P) → (∃𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑟 ∈ (1st ‘𝑧)) → 𝑥<P 𝑧))
40	39	ad2antrr 488	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥 ∈ P ∧ 𝑦 ∈ P ∧ 𝑧 ∈ P) ∧ 𝑟 ∈ (2nd ‘𝑥)) ∧ 𝑟 ∈ (1st ‘𝑧)) → (∃𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑟 ∈ (1st ‘𝑧)) → 𝑥<P 𝑧))
41	36, 40	mpd 13	. . . . . . . . . . . . . . 15 ⊢ ((((𝑥 ∈ P ∧ 𝑦 ∈ P ∧ 𝑧 ∈ P) ∧ 𝑟 ∈ (2nd ‘𝑥)) ∧ 𝑟 ∈ (1st ‘𝑧)) → 𝑥<P 𝑧)
42	41	ex 115	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P ∧ 𝑧 ∈ P) ∧ 𝑟 ∈ (2nd ‘𝑥)) → (𝑟 ∈ (1st ‘𝑧) → 𝑥<P 𝑧))
43	42	adantrr 479	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P ∧ 𝑧 ∈ P) ∧ (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑠 ∈ (1st ‘𝑦))) → (𝑟 ∈ (1st ‘𝑧) → 𝑥<P 𝑧))
44		elprnql 7541	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ ∈ P ∧ 𝑠 ∈ (1st ‘𝑦)) → 𝑠 ∈ Q)
45	7, 44	sylan 283	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ P ∧ 𝑠 ∈ (1st ‘𝑦)) → 𝑠 ∈ Q)
46		pm3.21 264	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑠 ∈ (1st ‘𝑦) → (𝑠 ∈ (2nd ‘𝑧) → (𝑠 ∈ (2nd ‘𝑧) ∧ 𝑠 ∈ (1st ‘𝑦))))
47	46	adantl 277	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ P ∧ 𝑠 ∈ (1st ‘𝑦)) → (𝑠 ∈ (2nd ‘𝑧) → (𝑠 ∈ (2nd ‘𝑧) ∧ 𝑠 ∈ (1st ‘𝑦))))
48		19.8a 1601	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑧) ∧ 𝑠 ∈ (1st ‘𝑦))) → ∃𝑠(𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑧) ∧ 𝑠 ∈ (1st ‘𝑦))))
49	45, 47, 48	syl6an 1445	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ P ∧ 𝑠 ∈ (1st ‘𝑦)) → (𝑠 ∈ (2nd ‘𝑧) → ∃𝑠(𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑧) ∧ 𝑠 ∈ (1st ‘𝑦)))))
50	49	3ad2antl2 1162	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P ∧ 𝑧 ∈ P) ∧ 𝑠 ∈ (1st ‘𝑦)) → (𝑠 ∈ (2nd ‘𝑧) → ∃𝑠(𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑧) ∧ 𝑠 ∈ (1st ‘𝑦)))))
51	50	imp 124	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑥 ∈ P ∧ 𝑦 ∈ P ∧ 𝑧 ∈ P) ∧ 𝑠 ∈ (1st ‘𝑦)) ∧ 𝑠 ∈ (2nd ‘𝑧)) → ∃𝑠(𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑧) ∧ 𝑠 ∈ (1st ‘𝑦))))
52		df-rex 2478	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑠 ∈ Q (𝑠 ∈ (2nd ‘𝑧) ∧ 𝑠 ∈ (1st ‘𝑦)) ↔ ∃𝑠(𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑧) ∧ 𝑠 ∈ (1st ‘𝑦))))
53	51, 52	sylibr 134	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥 ∈ P ∧ 𝑦 ∈ P ∧ 𝑧 ∈ P) ∧ 𝑠 ∈ (1st ‘𝑦)) ∧ 𝑠 ∈ (2nd ‘𝑧)) → ∃𝑠 ∈ Q (𝑠 ∈ (2nd ‘𝑧) ∧ 𝑠 ∈ (1st ‘𝑦)))
54		ltdfpr 7566	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑧 ∈ P ∧ 𝑦 ∈ P) → (𝑧<P 𝑦 ↔ ∃𝑠 ∈ Q (𝑠 ∈ (2nd ‘𝑧) ∧ 𝑠 ∈ (1st ‘𝑦))))
55	54	biimprd 158	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑧 ∈ P ∧ 𝑦 ∈ P) → (∃𝑠 ∈ Q (𝑠 ∈ (2nd ‘𝑧) ∧ 𝑠 ∈ (1st ‘𝑦)) → 𝑧<P 𝑦))
56	55	ancoms 268	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ P ∧ 𝑧 ∈ P) → (∃𝑠 ∈ Q (𝑠 ∈ (2nd ‘𝑧) ∧ 𝑠 ∈ (1st ‘𝑦)) → 𝑧<P 𝑦))
57	56	3adant1 1017	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P ∧ 𝑧 ∈ P) → (∃𝑠 ∈ Q (𝑠 ∈ (2nd ‘𝑧) ∧ 𝑠 ∈ (1st ‘𝑦)) → 𝑧<P 𝑦))
58	57	ad2antrr 488	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥 ∈ P ∧ 𝑦 ∈ P ∧ 𝑧 ∈ P) ∧ 𝑠 ∈ (1st ‘𝑦)) ∧ 𝑠 ∈ (2nd ‘𝑧)) → (∃𝑠 ∈ Q (𝑠 ∈ (2nd ‘𝑧) ∧ 𝑠 ∈ (1st ‘𝑦)) → 𝑧<P 𝑦))
59	53, 58	mpd 13	. . . . . . . . . . . . . . 15 ⊢ ((((𝑥 ∈ P ∧ 𝑦 ∈ P ∧ 𝑧 ∈ P) ∧ 𝑠 ∈ (1st ‘𝑦)) ∧ 𝑠 ∈ (2nd ‘𝑧)) → 𝑧<P 𝑦)
60	59	ex 115	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P ∧ 𝑧 ∈ P) ∧ 𝑠 ∈ (1st ‘𝑦)) → (𝑠 ∈ (2nd ‘𝑧) → 𝑧<P 𝑦))
61	60	adantrl 478	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P ∧ 𝑧 ∈ P) ∧ (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑠 ∈ (1st ‘𝑦))) → (𝑠 ∈ (2nd ‘𝑧) → 𝑧<P 𝑦))
62	43, 61	orim12d 787	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P ∧ 𝑧 ∈ P) ∧ (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑠 ∈ (1st ‘𝑦))) → ((𝑟 ∈ (1st ‘𝑧) ∨ 𝑠 ∈ (2nd ‘𝑧)) → (𝑥<P 𝑧 ∨ 𝑧<P 𝑦)))
63	62	adantlr 477	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ P ∧ 𝑦 ∈ P ∧ 𝑧 ∈ P) ∧ (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦))) ∧ (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑠 ∈ (1st ‘𝑦))) → ((𝑟 ∈ (1st ‘𝑧) ∨ 𝑠 ∈ (2nd ‘𝑧)) → (𝑥<P 𝑧 ∨ 𝑧<P 𝑦)))
64	63	adantr 276	. . . . . . . . . 10 ⊢ (((((𝑥 ∈ P ∧ 𝑦 ∈ P ∧ 𝑧 ∈ P) ∧ (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦))) ∧ (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑠 ∈ (1st ‘𝑦))) ∧ (𝑟 <Q 𝑞 ∧ 𝑞 <Q 𝑠)) → ((𝑟 ∈ (1st ‘𝑧) ∨ 𝑠 ∈ (2nd ‘𝑧)) → (𝑥<P 𝑧 ∨ 𝑧<P 𝑦)))
65	26, 64	mpd 13	. . . . . . . . 9 ⊢ (((((𝑥 ∈ P ∧ 𝑦 ∈ P ∧ 𝑧 ∈ P) ∧ (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦))) ∧ (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑠 ∈ (1st ‘𝑦))) ∧ (𝑟 <Q 𝑞 ∧ 𝑞 <Q 𝑠)) → (𝑥<P 𝑧 ∨ 𝑧<P 𝑦))
66	65	ex 115	. . . . . . . 8 ⊢ ((((𝑥 ∈ P ∧ 𝑦 ∈ P ∧ 𝑧 ∈ P) ∧ (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦))) ∧ (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑠 ∈ (1st ‘𝑦))) → ((𝑟 <Q 𝑞 ∧ 𝑞 <Q 𝑠) → (𝑥<P 𝑧 ∨ 𝑧<P 𝑦)))
67	66	rexlimdvva 2619	. . . . . . 7 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P ∧ 𝑧 ∈ P) ∧ (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦))) → (∃𝑟 ∈ (2nd ‘𝑥)∃𝑠 ∈ (1st ‘𝑦)(𝑟 <Q 𝑞 ∧ 𝑞 <Q 𝑠) → (𝑥<P 𝑧 ∨ 𝑧<P 𝑦)))
68	14, 67	mpd 13	. . . . . 6 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P ∧ 𝑧 ∈ P) ∧ (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦))) → (𝑥<P 𝑧 ∨ 𝑧<P 𝑦))
69	68	ex 115	. . . . 5 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P ∧ 𝑧 ∈ P) → ((𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦)) → (𝑥<P 𝑧 ∨ 𝑧<P 𝑦)))
70	69	rexlimdvw 2615	. . . 4 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P ∧ 𝑧 ∈ P) → (∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦)) → (𝑥<P 𝑧 ∨ 𝑧<P 𝑦)))
71	3, 70	sylbid 150	. . 3 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P ∧ 𝑧 ∈ P) → (𝑥<P 𝑦 → (𝑥<P 𝑧 ∨ 𝑧<P 𝑦)))
72	71	rgen3 2581	. 2 ⊢ ∀𝑥 ∈ P ∀𝑦 ∈ P ∀𝑧 ∈ P (𝑥<P 𝑦 → (𝑥<P 𝑧 ∨ 𝑧<P 𝑦))
73		df-iso 4328	. 2 ⊢ (<P Or P ↔ (<P Po P ∧ ∀𝑥 ∈ P ∀𝑦 ∈ P ∀𝑧 ∈ P (𝑥<P 𝑦 → (𝑥<P 𝑧 ∨ 𝑧<P 𝑦))))
74	1, 72, 73	mpbir2an 944	1 ⊢ <P Or P

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 709   ∧ w3a 980  ∃wex 1503   ∈ wcel 2164  ∀wral 2472  ∃wrex 2473  ⟨cop 3621   class class class wbr 4029   Po wpo 4325   Or wor 4326  ‘cfv 5254  1^st c1st 6191  2^nd c2nd 6192  Qcnq 7340   <_Q cltq 7345  Pcnp 7351  <_P cltp 7355

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-eprel 4320  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-oadd 6473  df-omul 6474  df-er 6587  df-ec 6589  df-qs 6593  df-ni 7364  df-mi 7366  df-lti 7367  df-enq 7407  df-nqqs 7408  df-ltnqqs 7413  df-inp 7526  df-iltp 7530

This theorem is referenced by:  prplnqu  7680  addextpr  7681  caucvgprprlemk  7743  caucvgprprlemnkltj  7749  caucvgprprlemnkeqj  7750  caucvgprprlemnjltk  7751  caucvgprprlemnbj  7753  caucvgprprlemml  7754  caucvgprprlemlol  7758  caucvgprprlemupu  7760  caucvgprprlemloc  7763  caucvgprprlemaddq  7768  suplocexprlemmu  7778  lttrsr  7822  ltposr  7823  ltsosr  7824  archsr  7842