Theorem sowlin 4085

Description: A strict order relation satisfies weak linearity. (Contributed by Jim Kingdon, 6-Oct-2018.)

Assertion

Ref	Expression
sowlin	⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐵𝑅𝐶 → (𝐵𝑅𝐷 ∨ 𝐷𝑅𝐶)))

Proof of Theorem sowlin

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

Step	Hyp	Ref	Expression
1		breq1 3795	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝑥𝑅𝑦 ↔ 𝐵𝑅𝑦))
2		breq1 3795	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝑥𝑅𝑧 ↔ 𝐵𝑅𝑧))
3	2	orbi1d 715	. . . . 5 ⊢ (𝑥 = 𝐵 → ((𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦) ↔ (𝐵𝑅𝑧 ∨ 𝑧𝑅𝑦)))
4	1, 3	imbi12d 227	. . . 4 ⊢ (𝑥 = 𝐵 → ((𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)) ↔ (𝐵𝑅𝑦 → (𝐵𝑅𝑧 ∨ 𝑧𝑅𝑦))))
5	4	imbi2d 223	. . 3 ⊢ (𝑥 = 𝐵 → ((𝑅 Or 𝐴 → (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦))) ↔ (𝑅 Or 𝐴 → (𝐵𝑅𝑦 → (𝐵𝑅𝑧 ∨ 𝑧𝑅𝑦)))))
6		breq2 3796	. . . . 5 ⊢ (𝑦 = 𝐶 → (𝐵𝑅𝑦 ↔ 𝐵𝑅𝐶))
7		breq2 3796	. . . . . 6 ⊢ (𝑦 = 𝐶 → (𝑧𝑅𝑦 ↔ 𝑧𝑅𝐶))
8	7	orbi2d 714	. . . . 5 ⊢ (𝑦 = 𝐶 → ((𝐵𝑅𝑧 ∨ 𝑧𝑅𝑦) ↔ (𝐵𝑅𝑧 ∨ 𝑧𝑅𝐶)))
9	6, 8	imbi12d 227	. . . 4 ⊢ (𝑦 = 𝐶 → ((𝐵𝑅𝑦 → (𝐵𝑅𝑧 ∨ 𝑧𝑅𝑦)) ↔ (𝐵𝑅𝐶 → (𝐵𝑅𝑧 ∨ 𝑧𝑅𝐶))))
10	9	imbi2d 223	. . 3 ⊢ (𝑦 = 𝐶 → ((𝑅 Or 𝐴 → (𝐵𝑅𝑦 → (𝐵𝑅𝑧 ∨ 𝑧𝑅𝑦))) ↔ (𝑅 Or 𝐴 → (𝐵𝑅𝐶 → (𝐵𝑅𝑧 ∨ 𝑧𝑅𝐶)))))
11		breq2 3796	. . . . . 6 ⊢ (𝑧 = 𝐷 → (𝐵𝑅𝑧 ↔ 𝐵𝑅𝐷))
12		breq1 3795	. . . . . 6 ⊢ (𝑧 = 𝐷 → (𝑧𝑅𝐶 ↔ 𝐷𝑅𝐶))
13	11, 12	orbi12d 717	. . . . 5 ⊢ (𝑧 = 𝐷 → ((𝐵𝑅𝑧 ∨ 𝑧𝑅𝐶) ↔ (𝐵𝑅𝐷 ∨ 𝐷𝑅𝐶)))
14	13	imbi2d 223	. . . 4 ⊢ (𝑧 = 𝐷 → ((𝐵𝑅𝐶 → (𝐵𝑅𝑧 ∨ 𝑧𝑅𝐶)) ↔ (𝐵𝑅𝐶 → (𝐵𝑅𝐷 ∨ 𝐷𝑅𝐶))))
15	14	imbi2d 223	. . 3 ⊢ (𝑧 = 𝐷 → ((𝑅 Or 𝐴 → (𝐵𝑅𝐶 → (𝐵𝑅𝑧 ∨ 𝑧𝑅𝐶))) ↔ (𝑅 Or 𝐴 → (𝐵𝑅𝐶 → (𝐵𝑅𝐷 ∨ 𝐷𝑅𝐶)))))
16		df-iso 4062	. . . . 5 ⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦))))
17		3anass 900	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) ↔ (𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)))
18		rsp 2386	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)) → (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦))))
19		rsp2 2388	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)) → ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦))))
20	18, 19	syl6 33	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)) → (𝑥 ∈ 𝐴 → ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)))))
21	20	impd 246	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)) → ((𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦))))
22	17, 21	syl5bi 145	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦))))
23	22	adantl 266	. . . . 5 ⊢ ((𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦))))
24	16, 23	sylbi 118	. . . 4 ⊢ (𝑅 Or 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦))))
25	24	com12 30	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑅 Or 𝐴 → (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦))))
26	5, 10, 15, 25	vtocl3ga 2640	. 2 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) → (𝑅 Or 𝐴 → (𝐵𝑅𝐶 → (𝐵𝑅𝐷 ∨ 𝐷𝑅𝐶))))
27	26	impcom 120	1 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐵𝑅𝐶 → (𝐵𝑅𝐷 ∨ 𝐷𝑅𝐶)))

Syntax hints:   → wi 4   ∧ wa 101   ∨ wo 639   ∧ w3a 896   = wceq 1259   ∈ wcel 1409  ∀wral 2323   class class class wbr 3792   Po wpo 4059   Or wor 4060

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038

This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576  df-un 2950  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-iso 4062

This theorem is referenced by:  sotri2  4750  sotri3  4751  addextpr  6777  cauappcvgprlemloc  6808  caucvgprlemloc  6831  caucvgprprlemloc  6859  caucvgprprlemaddq  6864  ltsosr  6907  axpre-ltwlin  7015  xrlelttr  8823  xrltletr  8824  xrletr  8825