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Theorem sowlin 4332
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 4018 . . . . 5 (𝑥 = 𝐵 → (𝑥𝑅𝑦𝐵𝑅𝑦))
2 breq1 4018 . . . . . 6 (𝑥 = 𝐵 → (𝑥𝑅𝑧𝐵𝑅𝑧))
32orbi1d 792 . . . . 5 (𝑥 = 𝐵 → ((𝑥𝑅𝑧𝑧𝑅𝑦) ↔ (𝐵𝑅𝑧𝑧𝑅𝑦)))
41, 3imbi12d 234 . . . 4 (𝑥 = 𝐵 → ((𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) ↔ (𝐵𝑅𝑦 → (𝐵𝑅𝑧𝑧𝑅𝑦))))
54imbi2d 230 . . 3 (𝑥 = 𝐵 → ((𝑅 Or 𝐴 → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))) ↔ (𝑅 Or 𝐴 → (𝐵𝑅𝑦 → (𝐵𝑅𝑧𝑧𝑅𝑦)))))
6 breq2 4019 . . . . 5 (𝑦 = 𝐶 → (𝐵𝑅𝑦𝐵𝑅𝐶))
7 breq2 4019 . . . . . 6 (𝑦 = 𝐶 → (𝑧𝑅𝑦𝑧𝑅𝐶))
87orbi2d 791 . . . . 5 (𝑦 = 𝐶 → ((𝐵𝑅𝑧𝑧𝑅𝑦) ↔ (𝐵𝑅𝑧𝑧𝑅𝐶)))
96, 8imbi12d 234 . . . 4 (𝑦 = 𝐶 → ((𝐵𝑅𝑦 → (𝐵𝑅𝑧𝑧𝑅𝑦)) ↔ (𝐵𝑅𝐶 → (𝐵𝑅𝑧𝑧𝑅𝐶))))
109imbi2d 230 . . 3 (𝑦 = 𝐶 → ((𝑅 Or 𝐴 → (𝐵𝑅𝑦 → (𝐵𝑅𝑧𝑧𝑅𝑦))) ↔ (𝑅 Or 𝐴 → (𝐵𝑅𝐶 → (𝐵𝑅𝑧𝑧𝑅𝐶)))))
11 breq2 4019 . . . . . 6 (𝑧 = 𝐷 → (𝐵𝑅𝑧𝐵𝑅𝐷))
12 breq1 4018 . . . . . 6 (𝑧 = 𝐷 → (𝑧𝑅𝐶𝐷𝑅𝐶))
1311, 12orbi12d 794 . . . . 5 (𝑧 = 𝐷 → ((𝐵𝑅𝑧𝑧𝑅𝐶) ↔ (𝐵𝑅𝐷𝐷𝑅𝐶)))
1413imbi2d 230 . . . 4 (𝑧 = 𝐷 → ((𝐵𝑅𝐶 → (𝐵𝑅𝑧𝑧𝑅𝐶)) ↔ (𝐵𝑅𝐶 → (𝐵𝑅𝐷𝐷𝑅𝐶))))
1514imbi2d 230 . . 3 (𝑧 = 𝐷 → ((𝑅 Or 𝐴 → (𝐵𝑅𝐶 → (𝐵𝑅𝑧𝑧𝑅𝐶))) ↔ (𝑅 Or 𝐴 → (𝐵𝑅𝐶 → (𝐵𝑅𝐷𝐷𝑅𝐶)))))
16 df-iso 4309 . . . . 5 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
17 3anass 983 . . . . . . 7 ((𝑥𝐴𝑦𝐴𝑧𝐴) ↔ (𝑥𝐴 ∧ (𝑦𝐴𝑧𝐴)))
18 rsp 2534 . . . . . . . . 9 (∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) → (𝑥𝐴 → ∀𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
19 rsp2 2537 . . . . . . . . 9 (∀𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) → ((𝑦𝐴𝑧𝐴) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
2018, 19syl6 33 . . . . . . . 8 (∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) → (𝑥𝐴 → ((𝑦𝐴𝑧𝐴) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)))))
2120impd 254 . . . . . . 7 (∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) → ((𝑥𝐴 ∧ (𝑦𝐴𝑧𝐴)) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
2217, 21biimtrid 152 . . . . . 6 (∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) → ((𝑥𝐴𝑦𝐴𝑧𝐴) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
2322adantl 277 . . . . 5 ((𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))) → ((𝑥𝐴𝑦𝐴𝑧𝐴) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
2416, 23sylbi 121 . . . 4 (𝑅 Or 𝐴 → ((𝑥𝐴𝑦𝐴𝑧𝐴) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
2524com12 30 . . 3 ((𝑥𝐴𝑦𝐴𝑧𝐴) → (𝑅 Or 𝐴 → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
265, 10, 15, 25vtocl3ga 2819 . 2 ((𝐵𝐴𝐶𝐴𝐷𝐴) → (𝑅 Or 𝐴 → (𝐵𝑅𝐶 → (𝐵𝑅𝐷𝐷𝑅𝐶))))
2726impcom 125 1 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → (𝐵𝑅𝐶 → (𝐵𝑅𝐷𝐷𝑅𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wo 709  w3a 979   = wceq 1363  wcel 2158  wral 2465   class class class wbr 4015   Po wpo 4306   Or wor 4307
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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-v 2751  df-un 3145  df-sn 3610  df-pr 3611  df-op 3613  df-br 4016  df-iso 4309
This theorem is referenced by:  sotri2  5038  sotri3  5039  suplub2ti  7014  addextpr  7634  cauappcvgprlemloc  7665  caucvgprlemloc  7688  caucvgprprlemloc  7716  caucvgprprlemaddq  7721  ltsosr  7777  suplocsrlem  7821  axpre-ltwlin  7896  xrlelttr  9820  xrltletr  9821  xrletr  9822  xrmaxiflemlub  11270
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