Theorem ltsopr 7799

Description: Positive real 'less than' is a weak linear order (in the sense of df-iso 4389). 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 7798	. 2 ⊢ <P Po P
2		ltdfpr 7709	. . . . 5 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → (𝑥<P 𝑦 ↔ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦))))
3	2	3adant3 1041	. . . 4 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P ∧ 𝑧 ∈ P) → (𝑥<P 𝑦 ↔ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦))))
4		prop 7678	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ P → ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ P)
5		prnminu 7692	. . . . . . . . . . . 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 7678	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ P → ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ ∈ P)
8		prnmaxl 7691	. . . . . . . . . . . 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 590	. . . . . . . . 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 2701	. . . . . . . . 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 1179	. . . . . . 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 7601	. . . . . . . . . . . . 13 ⊢ <Q Or Q
16		ltrelnq 7568	. . . . . . . . . . . . 13 ⊢ <Q ⊆ (Q × Q)
17	15, 16	sotri 5127	. . . . . . . . . . . 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 7678	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ P → ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∈ P)
20		prloc 7694	. . . . . . . . . . . . . . . 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 1185	. . . . . . . . . . . . . 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 7685	. . . . . . . . . . . . . . . . . . . . 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 1636	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑟 ∈ Q ∧ (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑟 ∈ (1st ‘𝑧))) → ∃𝑟(𝑟 ∈ Q ∧ (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑟 ∈ (1st ‘𝑧))))
32	28, 30, 31	syl6an 1476	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ P ∧ 𝑟 ∈ (2nd ‘𝑥)) → (𝑟 ∈ (1st ‘𝑧) → ∃𝑟(𝑟 ∈ Q ∧ (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑟 ∈ (1st ‘𝑧)))))
33	32	3ad2antl1 1183	. . . . . . . . . . . . . . . . . 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 2514	. . . . . . . . . . . . . . . . 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 7709	. . . . . . . . . . . . . . . . . . 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 1040	. . . . . . . . . . . . . . . . 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 7684	. . . . . . . . . . . . . . . . . . . . 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 1636	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑧) ∧ 𝑠 ∈ (1st ‘𝑦))) → ∃𝑠(𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑧) ∧ 𝑠 ∈ (1st ‘𝑦))))
49	45, 47, 48	syl6an 1476	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ P ∧ 𝑠 ∈ (1st ‘𝑦)) → (𝑠 ∈ (2nd ‘𝑧) → ∃𝑠(𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑧) ∧ 𝑠 ∈ (1st ‘𝑦)))))
50	49	3ad2antl2 1184	. . . . . . . . . . . . . . . . . 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 2514	. . . . . . . . . . . . . . . . 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 7709	. . . . . . . . . . . . . . . . . . . 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 1039	. . . . . . . . . . . . . . . . 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 791	. . . . . . . . . . . 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 2656	. . . . . . 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 2652	. . . 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 2617	. 2 ⊢ ∀𝑥 ∈ P ∀𝑦 ∈ P ∀𝑧 ∈ P (𝑥<P 𝑦 → (𝑥<P 𝑧 ∨ 𝑧<P 𝑦))
73		df-iso 4389	. 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 948	1 ⊢ <P Or P

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 713   ∧ w3a 1002  ∃wex 1538   ∈ wcel 2200  ∀wral 2508  ∃wrex 2509  ⟨cop 3669   class class class wbr 4083   Po wpo 4386   Or wor 4387  ‘cfv 5321  1^st c1st 6293  2^nd c2nd 6294  Qcnq 7483   <_Q cltq 7488  Pcnp 7494  <_P cltp 7498

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-setind 4630  ax-iinf 4681

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-eprel 4381  df-id 4385  df-po 4388  df-iso 4389  df-iord 4458  df-on 4460  df-suc 4463  df-iom 4684  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-f1 5326  df-fo 5327  df-f1o 5328  df-fv 5329  df-ov 6013  df-oprab 6014  df-mpo 6015  df-1st 6295  df-2nd 6296  df-recs 6462  df-irdg 6527  df-oadd 6577  df-omul 6578  df-er 6693  df-ec 6695  df-qs 6699  df-ni 7507  df-mi 7509  df-lti 7510  df-enq 7550  df-nqqs 7551  df-ltnqqs 7556  df-inp 7669  df-iltp 7673

This theorem is referenced by:  prplnqu  7823  addextpr  7824  caucvgprprlemk  7886  caucvgprprlemnkltj  7892  caucvgprprlemnkeqj  7893  caucvgprprlemnjltk  7894  caucvgprprlemnbj  7896  caucvgprprlemml  7897  caucvgprprlemlol  7901  caucvgprprlemupu  7903  caucvgprprlemloc  7906  caucvgprprlemaddq  7911  suplocexprlemmu  7921  lttrsr  7965  ltposr  7966  ltsosr  7967  archsr  7985