Theorem ltsopr 7927

Description: Positive real 'less than' is a weak linear order (in the sense of df-iso 4423). 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 7926	. 2 ⊢ <P Po P
2		ltdfpr 7837	. . . . 5 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → (𝑥<P 𝑦 ↔ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦))))
3	2	3adant3 1044	. . . 4 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P ∧ 𝑧 ∈ P) → (𝑥<P 𝑦 ↔ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦))))
4		prop 7806	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ P → ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ P)
5		prnminu 7820	. . . . . . . . . . . 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 7806	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ P → ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ ∈ P)
8		prnmaxl 7819	. . . . . . . . . . . 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 592	. . . . . . . . 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 2715	. . . . . . . . 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 1182	. . . . . . 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 7729	. . . . . . . . . . . . 13 ⊢ <Q Or Q
16		ltrelnq 7696	. . . . . . . . . . . . 13 ⊢ <Q ⊆ (Q × Q)
17	15, 16	sotri 5163	. . . . . . . . . . . 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 7806	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ P → ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∈ P)
20		prloc 7822	. . . . . . . . . . . . . . . 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 1188	. . . . . . . . . . . . . 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 7813	. . . . . . . . . . . . . . . . . . . . 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 1639	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑟 ∈ Q ∧ (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑟 ∈ (1st ‘𝑧))) → ∃𝑟(𝑟 ∈ Q ∧ (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑟 ∈ (1st ‘𝑧))))
32	28, 30, 31	syl6an 1479	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ P ∧ 𝑟 ∈ (2nd ‘𝑥)) → (𝑟 ∈ (1st ‘𝑧) → ∃𝑟(𝑟 ∈ Q ∧ (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑟 ∈ (1st ‘𝑧)))))
33	32	3ad2antl1 1186	. . . . . . . . . . . . . . . . . 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 2528	. . . . . . . . . . . . . . . . 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 7837	. . . . . . . . . . . . . . . . . . 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 1043	. . . . . . . . . . . . . . . . 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 7812	. . . . . . . . . . . . . . . . . . . . 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 1639	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑧) ∧ 𝑠 ∈ (1st ‘𝑦))) → ∃𝑠(𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑧) ∧ 𝑠 ∈ (1st ‘𝑦))))
49	45, 47, 48	syl6an 1479	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ P ∧ 𝑠 ∈ (1st ‘𝑦)) → (𝑠 ∈ (2nd ‘𝑧) → ∃𝑠(𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑧) ∧ 𝑠 ∈ (1st ‘𝑦)))))
50	49	3ad2antl2 1187	. . . . . . . . . . . . . . . . . 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 2528	. . . . . . . . . . . . . . . . 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 7837	. . . . . . . . . . . . . . . . . . . 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 1042	. . . . . . . . . . . . . . . . 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 794	. . . . . . . . . . . 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 2670	. . . . . . 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 2666	. . . 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 2631	. 2 ⊢ ∀𝑥 ∈ P ∀𝑦 ∈ P ∀𝑧 ∈ P (𝑥<P 𝑦 → (𝑥<P 𝑧 ∨ 𝑧<P 𝑦))
73		df-iso 4423	. 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 951	1 ⊢ <P Or P

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 716   ∧ w3a 1005  ∃wex 1541   ∈ wcel 2205  ∀wral 2522  ∃wrex 2523  ⟨cop 3697   class class class wbr 4114   Po wpo 4420   Or wor 4421  ‘cfv 5357  1^st c1st 6345  2^nd c2nd 6346  Qcnq 7611   <_Q cltq 7616  Pcnp 7622  <_P cltp 7626

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-eprel 4415  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-oadd 6664  df-omul 6665  df-er 6780  df-ec 6782  df-qs 6786  df-ni 7635  df-mi 7637  df-lti 7638  df-enq 7678  df-nqqs 7679  df-ltnqqs 7684  df-inp 7797  df-iltp 7801

This theorem is referenced by:  prplnqu  7951  addextpr  7952  caucvgprprlemk  8014  caucvgprprlemnkltj  8020  caucvgprprlemnkeqj  8021  caucvgprprlemnjltk  8022  caucvgprprlemnbj  8024  caucvgprprlemml  8025  caucvgprprlemlol  8029  caucvgprprlemupu  8031  caucvgprprlemloc  8034  caucvgprprlemaddq  8039  suplocexprlemmu  8049  lttrsr  8093  ltposr  8094  ltsosr  8095  archsr  8113