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Theorem sotr3 33249
 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 1135 . . . . . . 7 ((𝑋𝐴𝑌𝐴𝑍𝐴) → 𝑍𝐴)
2 simp2 1134 . . . . . . 7 ((𝑋𝐴𝑌𝐴𝑍𝐴) → 𝑌𝐴)
31, 2jca 515 . . . . . 6 ((𝑋𝐴𝑌𝐴𝑍𝐴) → (𝑍𝐴𝑌𝐴))
4 sotric 5470 . . . . . 6 ((𝑅 Or 𝐴 ∧ (𝑍𝐴𝑌𝐴)) → (𝑍𝑅𝑌 ↔ ¬ (𝑍 = 𝑌𝑌𝑅𝑍)))
53, 4sylan2 595 . . . . 5 ((𝑅 Or 𝐴 ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) → (𝑍𝑅𝑌 ↔ ¬ (𝑍 = 𝑌𝑌𝑅𝑍)))
65con2bid 358 . . . 4 ((𝑅 Or 𝐴 ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) → ((𝑍 = 𝑌𝑌𝑅𝑍) ↔ ¬ 𝑍𝑅𝑌))
76adantr 484 . . 3 (((𝑅 Or 𝐴 ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) ∧ 𝑋𝑅𝑌) → ((𝑍 = 𝑌𝑌𝑅𝑍) ↔ ¬ 𝑍𝑅𝑌))
8 breq2 5036 . . . . . 6 (𝑍 = 𝑌 → (𝑋𝑅𝑍𝑋𝑅𝑌))
98biimprcd 253 . . . . 5 (𝑋𝑅𝑌 → (𝑍 = 𝑌𝑋𝑅𝑍))
109adantl 485 . . . 4 (((𝑅 Or 𝐴 ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) ∧ 𝑋𝑅𝑌) → (𝑍 = 𝑌𝑋𝑅𝑍))
11 sotr 5466 . . . . 5 ((𝑅 Or 𝐴 ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) → ((𝑋𝑅𝑌𝑌𝑅𝑍) → 𝑋𝑅𝑍))
1211expdimp 456 . . . 4 (((𝑅 Or 𝐴 ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) ∧ 𝑋𝑅𝑌) → (𝑌𝑅𝑍𝑋𝑅𝑍))
1310, 12jaod 856 . . 3 (((𝑅 Or 𝐴 ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) ∧ 𝑋𝑅𝑌) → ((𝑍 = 𝑌𝑌𝑅𝑍) → 𝑋𝑅𝑍))
147, 13sylbird 263 . 2 (((𝑅 Or 𝐴 ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) ∧ 𝑋𝑅𝑌) → (¬ 𝑍𝑅𝑌𝑋𝑅𝑍))
1514expimpd 457 1 ((𝑅 Or 𝐴 ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) → ((𝑋𝑅𝑌 ∧ ¬ 𝑍𝑅𝑌) → 𝑋𝑅𝑍))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   class class class wbr 5032   Or wor 5442 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-v 3411  df-un 3863  df-sn 4523  df-pr 4525  df-op 4529  df-br 5033  df-po 5443  df-so 5444 This theorem is referenced by:  nosupbnd2  33484  noinfbnd1  33497  sltletr  33524
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