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Theorem sowlin 4237
 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 3927 . . . . 5 (𝑥 = 𝐵 → (𝑥𝑅𝑦𝐵𝑅𝑦))
2 breq1 3927 . . . . . 6 (𝑥 = 𝐵 → (𝑥𝑅𝑧𝐵𝑅𝑧))
32orbi1d 780 . . . . 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 3928 . . . . 5 (𝑦 = 𝐶 → (𝐵𝑅𝑦𝐵𝑅𝐶))
7 breq2 3928 . . . . . 6 (𝑦 = 𝐶 → (𝑧𝑅𝑦𝑧𝑅𝐶))
87orbi2d 779 . . . . 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 3928 . . . . . 6 (𝑧 = 𝐷 → (𝐵𝑅𝑧𝐵𝑅𝐷))
12 breq1 3927 . . . . . 6 (𝑧 = 𝐷 → (𝑧𝑅𝐶𝐷𝑅𝐶))
1311, 12orbi12d 782 . . . . 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 4214 . . . . 5 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
17 3anass 966 . . . . . . 7 ((𝑥𝐴𝑦𝐴𝑧𝐴) ↔ (𝑥𝐴 ∧ (𝑦𝐴𝑧𝐴)))
18 rsp 2478 . . . . . . . . 9 (∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) → (𝑥𝐴 → ∀𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
19 rsp2 2480 . . . . . . . . 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 275 . . . . 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 2751 . 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 697   ∧ w3a 962   = wceq 1331   ∈ wcel 1480  ∀wral 2414   class class class wbr 3924   Po wpo 4211   Or wor 4212 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-v 2683  df-un 3070  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-iso 4214 This theorem is referenced by:  sotri2  4931  sotri3  4932  suplub2ti  6881  addextpr  7422  cauappcvgprlemloc  7453  caucvgprlemloc  7476  caucvgprprlemloc  7504  caucvgprprlemaddq  7509  ltsosr  7565  suplocsrlem  7609  axpre-ltwlin  7684  xrlelttr  9582  xrltletr  9583  xrletr  9584  xrmaxiflemlub  11010
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