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Theorem sotr3 5632
Description: Transitivity law for strict orderings. (Contributed by Scott Fenton, 24-Nov-2021.)
Assertion
Ref Expression
sotr3 ((𝑅 Or 𝐴 ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) → ((𝑋𝑅𝑌 ∧ ¬ 𝑍𝑅𝑌) → 𝑋𝑅𝑍))

Proof of Theorem sotr3
StepHypRef Expression
1 simp3 1138 . . . . . . 7 ((𝑋𝐴𝑌𝐴𝑍𝐴) → 𝑍𝐴)
2 simp2 1137 . . . . . . 7 ((𝑋𝐴𝑌𝐴𝑍𝐴) → 𝑌𝐴)
31, 2jca 511 . . . . . 6 ((𝑋𝐴𝑌𝐴𝑍𝐴) → (𝑍𝐴𝑌𝐴))
4 sotric 5621 . . . . . 6 ((𝑅 Or 𝐴 ∧ (𝑍𝐴𝑌𝐴)) → (𝑍𝑅𝑌 ↔ ¬ (𝑍 = 𝑌𝑌𝑅𝑍)))
53, 4sylan2 593 . . . . 5 ((𝑅 Or 𝐴 ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) → (𝑍𝑅𝑌 ↔ ¬ (𝑍 = 𝑌𝑌𝑅𝑍)))
65con2bid 354 . . . 4 ((𝑅 Or 𝐴 ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) → ((𝑍 = 𝑌𝑌𝑅𝑍) ↔ ¬ 𝑍𝑅𝑌))
76adantr 480 . . 3 (((𝑅 Or 𝐴 ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) ∧ 𝑋𝑅𝑌) → ((𝑍 = 𝑌𝑌𝑅𝑍) ↔ ¬ 𝑍𝑅𝑌))
8 breq2 5146 . . . . . 6 (𝑍 = 𝑌 → (𝑋𝑅𝑍𝑋𝑅𝑌))
98biimprcd 250 . . . . 5 (𝑋𝑅𝑌 → (𝑍 = 𝑌𝑋𝑅𝑍))
109adantl 481 . . . 4 (((𝑅 Or 𝐴 ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) ∧ 𝑋𝑅𝑌) → (𝑍 = 𝑌𝑋𝑅𝑍))
11 sotr 5616 . . . . 5 ((𝑅 Or 𝐴 ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) → ((𝑋𝑅𝑌𝑌𝑅𝑍) → 𝑋𝑅𝑍))
1211expdimp 452 . . . 4 (((𝑅 Or 𝐴 ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) ∧ 𝑋𝑅𝑌) → (𝑌𝑅𝑍𝑋𝑅𝑍))
1310, 12jaod 859 . . 3 (((𝑅 Or 𝐴 ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) ∧ 𝑋𝑅𝑌) → ((𝑍 = 𝑌𝑌𝑅𝑍) → 𝑋𝑅𝑍))
147, 13sylbird 260 . 2 (((𝑅 Or 𝐴 ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) ∧ 𝑋𝑅𝑌) → (¬ 𝑍𝑅𝑌𝑋𝑅𝑍))
1514expimpd 453 1 ((𝑅 Or 𝐴 ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) → ((𝑋𝑅𝑌 ∧ ¬ 𝑍𝑅𝑌) → 𝑋𝑅𝑍))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847  w3a 1086   = wceq 1539  wcel 2107   class class class wbr 5142   Or wor 5590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-po 5591  df-so 5592
This theorem is referenced by:  nosupbnd2  27762  noinfbnd1  27775  sltletr  27802
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