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Theorem sowlin 4171
Description: A strict order relation satisfies weak linearity. (Contributed by Jim Kingdon, 6-Oct-2018.)
Assertion
Ref Expression
sowlin ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → (𝐵𝑅𝐶 → (𝐵𝑅𝐷𝐷𝑅𝐶)))

Proof of Theorem sowlin
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq1 3870 . . . . 5 (𝑥 = 𝐵 → (𝑥𝑅𝑦𝐵𝑅𝑦))
2 breq1 3870 . . . . . 6 (𝑥 = 𝐵 → (𝑥𝑅𝑧𝐵𝑅𝑧))
32orbi1d 743 . . . . 5 (𝑥 = 𝐵 → ((𝑥𝑅𝑧𝑧𝑅𝑦) ↔ (𝐵𝑅𝑧𝑧𝑅𝑦)))
41, 3imbi12d 233 . . . 4 (𝑥 = 𝐵 → ((𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) ↔ (𝐵𝑅𝑦 → (𝐵𝑅𝑧𝑧𝑅𝑦))))
54imbi2d 229 . . 3 (𝑥 = 𝐵 → ((𝑅 Or 𝐴 → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))) ↔ (𝑅 Or 𝐴 → (𝐵𝑅𝑦 → (𝐵𝑅𝑧𝑧𝑅𝑦)))))
6 breq2 3871 . . . . 5 (𝑦 = 𝐶 → (𝐵𝑅𝑦𝐵𝑅𝐶))
7 breq2 3871 . . . . . 6 (𝑦 = 𝐶 → (𝑧𝑅𝑦𝑧𝑅𝐶))
87orbi2d 742 . . . . 5 (𝑦 = 𝐶 → ((𝐵𝑅𝑧𝑧𝑅𝑦) ↔ (𝐵𝑅𝑧𝑧𝑅𝐶)))
96, 8imbi12d 233 . . . 4 (𝑦 = 𝐶 → ((𝐵𝑅𝑦 → (𝐵𝑅𝑧𝑧𝑅𝑦)) ↔ (𝐵𝑅𝐶 → (𝐵𝑅𝑧𝑧𝑅𝐶))))
109imbi2d 229 . . 3 (𝑦 = 𝐶 → ((𝑅 Or 𝐴 → (𝐵𝑅𝑦 → (𝐵𝑅𝑧𝑧𝑅𝑦))) ↔ (𝑅 Or 𝐴 → (𝐵𝑅𝐶 → (𝐵𝑅𝑧𝑧𝑅𝐶)))))
11 breq2 3871 . . . . . 6 (𝑧 = 𝐷 → (𝐵𝑅𝑧𝐵𝑅𝐷))
12 breq1 3870 . . . . . 6 (𝑧 = 𝐷 → (𝑧𝑅𝐶𝐷𝑅𝐶))
1311, 12orbi12d 745 . . . . 5 (𝑧 = 𝐷 → ((𝐵𝑅𝑧𝑧𝑅𝐶) ↔ (𝐵𝑅𝐷𝐷𝑅𝐶)))
1413imbi2d 229 . . . 4 (𝑧 = 𝐷 → ((𝐵𝑅𝐶 → (𝐵𝑅𝑧𝑧𝑅𝐶)) ↔ (𝐵𝑅𝐶 → (𝐵𝑅𝐷𝐷𝑅𝐶))))
1514imbi2d 229 . . 3 (𝑧 = 𝐷 → ((𝑅 Or 𝐴 → (𝐵𝑅𝐶 → (𝐵𝑅𝑧𝑧𝑅𝐶))) ↔ (𝑅 Or 𝐴 → (𝐵𝑅𝐶 → (𝐵𝑅𝐷𝐷𝑅𝐶)))))
16 df-iso 4148 . . . . 5 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
17 3anass 931 . . . . . . 7 ((𝑥𝐴𝑦𝐴𝑧𝐴) ↔ (𝑥𝐴 ∧ (𝑦𝐴𝑧𝐴)))
18 rsp 2434 . . . . . . . . 9 (∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) → (𝑥𝐴 → ∀𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
19 rsp2 2436 . . . . . . . . 9 (∀𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) → ((𝑦𝐴𝑧𝐴) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
2018, 19syl6 33 . . . . . . . 8 (∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) → (𝑥𝐴 → ((𝑦𝐴𝑧𝐴) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)))))
2120impd 252 . . . . . . 7 (∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) → ((𝑥𝐴 ∧ (𝑦𝐴𝑧𝐴)) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
2217, 21syl5bi 151 . . . . . 6 (∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) → ((𝑥𝐴𝑦𝐴𝑧𝐴) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
2322adantl 272 . . . . 5 ((𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))) → ((𝑥𝐴𝑦𝐴𝑧𝐴) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
2416, 23sylbi 120 . . . 4 (𝑅 Or 𝐴 → ((𝑥𝐴𝑦𝐴𝑧𝐴) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
2524com12 30 . . 3 ((𝑥𝐴𝑦𝐴𝑧𝐴) → (𝑅 Or 𝐴 → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
265, 10, 15, 25vtocl3ga 2703 . 2 ((𝐵𝐴𝐶𝐴𝐷𝐴) → (𝑅 Or 𝐴 → (𝐵𝑅𝐶 → (𝐵𝑅𝐷𝐷𝑅𝐶))))
2726impcom 124 1 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → (𝐵𝑅𝐶 → (𝐵𝑅𝐷𝐷𝑅𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wo 667  w3a 927   = wceq 1296  wcel 1445  wral 2370   class class class wbr 3867   Po wpo 4145   Or wor 4146
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-v 2635  df-un 3017  df-sn 3472  df-pr 3473  df-op 3475  df-br 3868  df-iso 4148
This theorem is referenced by:  sotri2  4862  sotri3  4863  suplub2ti  6776  addextpr  7277  cauappcvgprlemloc  7308  caucvgprlemloc  7331  caucvgprprlemloc  7359  caucvgprprlemaddq  7364  ltsosr  7407  axpre-ltwlin  7515  xrlelttr  9372  xrltletr  9373  xrletr  9374  xrmaxiflemlub  10807
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