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Theorem sotr3 5637
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 1137 . . . . . . 7 ((𝑋𝐴𝑌𝐴𝑍𝐴) → 𝑍𝐴)
2 simp2 1136 . . . . . . 7 ((𝑋𝐴𝑌𝐴𝑍𝐴) → 𝑌𝐴)
31, 2jca 511 . . . . . 6 ((𝑋𝐴𝑌𝐴𝑍𝐴) → (𝑍𝐴𝑌𝐴))
4 sotric 5626 . . . . . 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 5152 . . . . . 6 (𝑍 = 𝑌 → (𝑋𝑅𝑍𝑋𝑅𝑌))
98biimprcd 250 . . . . 5 (𝑋𝑅𝑌 → (𝑍 = 𝑌𝑋𝑅𝑍))
109adantl 481 . . . 4 (((𝑅 Or 𝐴 ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) ∧ 𝑋𝑅𝑌) → (𝑍 = 𝑌𝑋𝑅𝑍))
11 sotr 5622 . . . . 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 1537  wcel 2106   class class class wbr 5148   Or wor 5596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-po 5597  df-so 5598
This theorem is referenced by:  nosupbnd2  27776  noinfbnd1  27789  sltletr  27816
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