Theorem sowlin 4417

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 4091	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝑥𝑅𝑦 ↔ 𝐵𝑅𝑦))
2		breq1 4091	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝑥𝑅𝑧 ↔ 𝐵𝑅𝑧))
3	2	orbi1d 798	. . . . 5 ⊢ (𝑥 = 𝐵 → ((𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦) ↔ (𝐵𝑅𝑧 ∨ 𝑧𝑅𝑦)))
4	1, 3	imbi12d 234	. . . 4 ⊢ (𝑥 = 𝐵 → ((𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)) ↔ (𝐵𝑅𝑦 → (𝐵𝑅𝑧 ∨ 𝑧𝑅𝑦))))
5	4	imbi2d 230	. . 3 ⊢ (𝑥 = 𝐵 → ((𝑅 Or 𝐴 → (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦))) ↔ (𝑅 Or 𝐴 → (𝐵𝑅𝑦 → (𝐵𝑅𝑧 ∨ 𝑧𝑅𝑦)))))
6		breq2 4092	. . . . 5 ⊢ (𝑦 = 𝐶 → (𝐵𝑅𝑦 ↔ 𝐵𝑅𝐶))
7		breq2 4092	. . . . . 6 ⊢ (𝑦 = 𝐶 → (𝑧𝑅𝑦 ↔ 𝑧𝑅𝐶))
8	7	orbi2d 797	. . . . 5 ⊢ (𝑦 = 𝐶 → ((𝐵𝑅𝑧 ∨ 𝑧𝑅𝑦) ↔ (𝐵𝑅𝑧 ∨ 𝑧𝑅𝐶)))
9	6, 8	imbi12d 234	. . . 4 ⊢ (𝑦 = 𝐶 → ((𝐵𝑅𝑦 → (𝐵𝑅𝑧 ∨ 𝑧𝑅𝑦)) ↔ (𝐵𝑅𝐶 → (𝐵𝑅𝑧 ∨ 𝑧𝑅𝐶))))
10	9	imbi2d 230	. . 3 ⊢ (𝑦 = 𝐶 → ((𝑅 Or 𝐴 → (𝐵𝑅𝑦 → (𝐵𝑅𝑧 ∨ 𝑧𝑅𝑦))) ↔ (𝑅 Or 𝐴 → (𝐵𝑅𝐶 → (𝐵𝑅𝑧 ∨ 𝑧𝑅𝐶)))))
11		breq2 4092	. . . . . 6 ⊢ (𝑧 = 𝐷 → (𝐵𝑅𝑧 ↔ 𝐵𝑅𝐷))
12		breq1 4091	. . . . . 6 ⊢ (𝑧 = 𝐷 → (𝑧𝑅𝐶 ↔ 𝐷𝑅𝐶))
13	11, 12	orbi12d 800	. . . . 5 ⊢ (𝑧 = 𝐷 → ((𝐵𝑅𝑧 ∨ 𝑧𝑅𝐶) ↔ (𝐵𝑅𝐷 ∨ 𝐷𝑅𝐶)))
14	13	imbi2d 230	. . . 4 ⊢ (𝑧 = 𝐷 → ((𝐵𝑅𝐶 → (𝐵𝑅𝑧 ∨ 𝑧𝑅𝐶)) ↔ (𝐵𝑅𝐶 → (𝐵𝑅𝐷 ∨ 𝐷𝑅𝐶))))
15	14	imbi2d 230	. . 3 ⊢ (𝑧 = 𝐷 → ((𝑅 Or 𝐴 → (𝐵𝑅𝐶 → (𝐵𝑅𝑧 ∨ 𝑧𝑅𝐶))) ↔ (𝑅 Or 𝐴 → (𝐵𝑅𝐶 → (𝐵𝑅𝐷 ∨ 𝐷𝑅𝐶)))))
16		df-iso 4394	. . . . 5 ⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦))))
17		3anass 1008	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) ↔ (𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)))
18		rsp 2579	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)) → (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦))))
19		rsp2 2582	. . . . . . . . 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 254	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)) → ((𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦))))
22	17, 21	biimtrid 152	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦))))
23	22	adantl 277	. . . . 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 121	. . . 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 2874	. 2 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) → (𝑅 Or 𝐴 → (𝐵𝑅𝐶 → (𝐵𝑅𝐷 ∨ 𝐷𝑅𝐶))))
27	26	impcom 125	1 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐵𝑅𝐶 → (𝐵𝑅𝐷 ∨ 𝐷𝑅𝐶)))

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 715   ∧ w3a 1004   = wceq 1397   ∈ wcel 2202  ∀wral 2510   class class class wbr 4088   Po wpo 4391   Or wor 4392

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-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-iso 4394

This theorem is referenced by:  sotri2  5134  sotri3  5135  suplub2ti  7200  addextpr  7841  cauappcvgprlemloc  7872  caucvgprlemloc  7895  caucvgprprlemloc  7923  caucvgprprlemaddq  7928  ltsosr  7984  suplocsrlem  8028  axpre-ltwlin  8103  xrlelttr  10041  xrltletr  10042  xrletr  10043  xrmaxiflemlub  11810