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Theorem sowlin 4121
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 3823 . . . . 5 (𝑥 = 𝐵 → (𝑥𝑅𝑦𝐵𝑅𝑦))
2 breq1 3823 . . . . . 6 (𝑥 = 𝐵 → (𝑥𝑅𝑧𝐵𝑅𝑧))
32orbi1d 738 . . . . 5 (𝑥 = 𝐵 → ((𝑥𝑅𝑧𝑧𝑅𝑦) ↔ (𝐵𝑅𝑧𝑧𝑅𝑦)))
41, 3imbi12d 232 . . . 4 (𝑥 = 𝐵 → ((𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) ↔ (𝐵𝑅𝑦 → (𝐵𝑅𝑧𝑧𝑅𝑦))))
54imbi2d 228 . . 3 (𝑥 = 𝐵 → ((𝑅 Or 𝐴 → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))) ↔ (𝑅 Or 𝐴 → (𝐵𝑅𝑦 → (𝐵𝑅𝑧𝑧𝑅𝑦)))))
6 breq2 3824 . . . . 5 (𝑦 = 𝐶 → (𝐵𝑅𝑦𝐵𝑅𝐶))
7 breq2 3824 . . . . . 6 (𝑦 = 𝐶 → (𝑧𝑅𝑦𝑧𝑅𝐶))
87orbi2d 737 . . . . 5 (𝑦 = 𝐶 → ((𝐵𝑅𝑧𝑧𝑅𝑦) ↔ (𝐵𝑅𝑧𝑧𝑅𝐶)))
96, 8imbi12d 232 . . . 4 (𝑦 = 𝐶 → ((𝐵𝑅𝑦 → (𝐵𝑅𝑧𝑧𝑅𝑦)) ↔ (𝐵𝑅𝐶 → (𝐵𝑅𝑧𝑧𝑅𝐶))))
109imbi2d 228 . . 3 (𝑦 = 𝐶 → ((𝑅 Or 𝐴 → (𝐵𝑅𝑦 → (𝐵𝑅𝑧𝑧𝑅𝑦))) ↔ (𝑅 Or 𝐴 → (𝐵𝑅𝐶 → (𝐵𝑅𝑧𝑧𝑅𝐶)))))
11 breq2 3824 . . . . . 6 (𝑧 = 𝐷 → (𝐵𝑅𝑧𝐵𝑅𝐷))
12 breq1 3823 . . . . . 6 (𝑧 = 𝐷 → (𝑧𝑅𝐶𝐷𝑅𝐶))
1311, 12orbi12d 740 . . . . 5 (𝑧 = 𝐷 → ((𝐵𝑅𝑧𝑧𝑅𝐶) ↔ (𝐵𝑅𝐷𝐷𝑅𝐶)))
1413imbi2d 228 . . . 4 (𝑧 = 𝐷 → ((𝐵𝑅𝐶 → (𝐵𝑅𝑧𝑧𝑅𝐶)) ↔ (𝐵𝑅𝐶 → (𝐵𝑅𝐷𝐷𝑅𝐶))))
1514imbi2d 228 . . 3 (𝑧 = 𝐷 → ((𝑅 Or 𝐴 → (𝐵𝑅𝐶 → (𝐵𝑅𝑧𝑧𝑅𝐶))) ↔ (𝑅 Or 𝐴 → (𝐵𝑅𝐶 → (𝐵𝑅𝐷𝐷𝑅𝐶)))))
16 df-iso 4098 . . . . 5 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
17 3anass 926 . . . . . . 7 ((𝑥𝐴𝑦𝐴𝑧𝐴) ↔ (𝑥𝐴 ∧ (𝑦𝐴𝑧𝐴)))
18 rsp 2419 . . . . . . . . 9 (∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) → (𝑥𝐴 → ∀𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
19 rsp2 2421 . . . . . . . . 9 (∀𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) → ((𝑦𝐴𝑧𝐴) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
2018, 19syl6 33 . . . . . . . 8 (∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) → (𝑥𝐴 → ((𝑦𝐴𝑧𝐴) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)))))
2120impd 251 . . . . . . 7 (∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) → ((𝑥𝐴 ∧ (𝑦𝐴𝑧𝐴)) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
2217, 21syl5bi 150 . . . . . 6 (∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) → ((𝑥𝐴𝑦𝐴𝑧𝐴) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
2322adantl 271 . . . . 5 ((𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))) → ((𝑥𝐴𝑦𝐴𝑧𝐴) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
2416, 23sylbi 119 . . . 4 (𝑅 Or 𝐴 → ((𝑥𝐴𝑦𝐴𝑧𝐴) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
2524com12 30 . . 3 ((𝑥𝐴𝑦𝐴𝑧𝐴) → (𝑅 Or 𝐴 → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
265, 10, 15, 25vtocl3ga 2682 . 2 ((𝐵𝐴𝐶𝐴𝐷𝐴) → (𝑅 Or 𝐴 → (𝐵𝑅𝐶 → (𝐵𝑅𝐷𝐷𝑅𝐶))))
2726impcom 123 1 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → (𝐵𝑅𝐶 → (𝐵𝑅𝐷𝐷𝑅𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wo 662  w3a 922   = wceq 1287  wcel 1436  wral 2355   class class class wbr 3820   Po wpo 4095   Or wor 4096
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-v 2617  df-un 2992  df-sn 3437  df-pr 3438  df-op 3440  df-br 3821  df-iso 4098
This theorem is referenced by:  sotri2  4796  sotri3  4797  suplub2ti  6640  addextpr  7124  cauappcvgprlemloc  7155  caucvgprlemloc  7178  caucvgprprlemloc  7206  caucvgprprlemaddq  7211  ltsosr  7254  axpre-ltwlin  7362  xrlelttr  9203  xrltletr  9204  xrletr  9205
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