Theorem ltsopr 7957

Description: Positive real 'less than' is a weak linear order (in the sense of df-iso 4440). 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 7956	. 2 ⊢ <P Po P
2		ltdfpr 7867	. . . . 5 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → (𝑥<P 𝑦 ↔ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦))))
3	2	3adant3 1048	. . . 4 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P ∧ 𝑧 ∈ P) → (𝑥<P 𝑦 ↔ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦))))
4		prop 7836	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ P → ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ P)
5		prnminu 7850	. . . . . . . . . . . 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 7836	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ P → ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ ∈ P)
8		prnmaxl 7849	. . . . . . . . . . . 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 596	. . . . . . . . 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 2721	. . . . . . . . 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 1186	. . . . . . 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 7759	. . . . . . . . . . . . 13 ⊢ <Q Or Q
16		ltrelnq 7726	. . . . . . . . . . . . 13 ⊢ <Q ⊆ (Q × Q)
17	15, 16	sotri 5181	. . . . . . . . . . . 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 7836	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ P → ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∈ P)
20		prloc 7852	. . . . . . . . . . . . . . . 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 1192	. . . . . . . . . . . . . 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 492	. . . . . . . . . . 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 7843	. . . . . . . . . . . . . . . . . . . . 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 1643	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑟 ∈ Q ∧ (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑟 ∈ (1st ‘𝑧))) → ∃𝑟(𝑟 ∈ Q ∧ (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑟 ∈ (1st ‘𝑧))))
32	28, 30, 31	syl6an 1483	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ P ∧ 𝑟 ∈ (2nd ‘𝑥)) → (𝑟 ∈ (1st ‘𝑧) → ∃𝑟(𝑟 ∈ Q ∧ (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑟 ∈ (1st ‘𝑧)))))
33	32	3ad2antl1 1190	. . . . . . . . . . . . . . . . . 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 2534	. . . . . . . . . . . . . . . . 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 7867	. . . . . . . . . . . . . . . . . . 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 1047	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P ∧ 𝑧 ∈ P) → (∃𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑟 ∈ (1st ‘𝑧)) → 𝑥<P 𝑧))
40	39	ad2antrr 492	. . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P ∧ 𝑧 ∈ P) ∧ (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑠 ∈ (1st ‘𝑦))) → (𝑟 ∈ (1st ‘𝑧) → 𝑥<P 𝑧))
44		elprnql 7842	. . . . . . . . . . . . . . . . . . . . 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 1643	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑧) ∧ 𝑠 ∈ (1st ‘𝑦))) → ∃𝑠(𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑧) ∧ 𝑠 ∈ (1st ‘𝑦))))
49	45, 47, 48	syl6an 1483	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ P ∧ 𝑠 ∈ (1st ‘𝑦)) → (𝑠 ∈ (2nd ‘𝑧) → ∃𝑠(𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑧) ∧ 𝑠 ∈ (1st ‘𝑦)))))
50	49	3ad2antl2 1191	. . . . . . . . . . . . . . . . . 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 2534	. . . . . . . . . . . . . . . . 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 7867	. . . . . . . . . . . . . . . . . . . 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 1046	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P ∧ 𝑧 ∈ P) → (∃𝑠 ∈ Q (𝑠 ∈ (2nd ‘𝑧) ∧ 𝑠 ∈ (1st ‘𝑦)) → 𝑧<P 𝑦))
58	57	ad2antrr 492	. . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P ∧ 𝑧 ∈ P) ∧ (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑠 ∈ (1st ‘𝑦))) → (𝑠 ∈ (2nd ‘𝑧) → 𝑧<P 𝑦))
62	43, 61	orim12d 798	. . . . . . . . . . . 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 481	. . . . . . . . . . 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 2676	. . . . . . 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 2672	. . . 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 2637	. 2 ⊢ ∀𝑥 ∈ P ∀𝑦 ∈ P ∀𝑧 ∈ P (𝑥<P 𝑦 → (𝑥<P 𝑧 ∨ 𝑧<P 𝑦))
73		df-iso 4440	. 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 955	1 ⊢ <P Or P

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 720   ∧ w3a 1009  ∃wex 1545   ∈ wcel 2209  ∀wral 2528  ∃wrex 2529  ⟨cop 3711   class class class wbr 4128   Po wpo 4437   Or wor 4438  ‘cfv 5375  1^st c1st 6366  2^nd c2nd 6367  Qcnq 7641   <_Q cltq 7646  Pcnp 7652  <_P cltp 7656

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-eprel 4432  df-id 4436  df-po 4439  df-iso 4440  df-iord 4509  df-on 4511  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-irdg 6635  df-oadd 6685  df-omul 6686  df-er 6801  df-ec 6803  df-qs 6807  df-ni 7665  df-mi 7667  df-lti 7668  df-enq 7708  df-nqqs 7709  df-ltnqqs 7714  df-inp 7827  df-iltp 7831

This theorem is referenced by:  prplnqu  7981  addextpr  7982  caucvgprprlemk  8044  caucvgprprlemnkltj  8050  caucvgprprlemnkeqj  8051  caucvgprprlemnjltk  8052  caucvgprprlemnbj  8054  caucvgprprlemml  8055  caucvgprprlemlol  8059  caucvgprprlemupu  8061  caucvgprprlemloc  8064  caucvgprprlemaddq  8069  suplocexprlemmu  8079  lttrsr  8123  ltposr  8124  ltsosr  8125  archsr  8143