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Theorem sotr3 31959
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 1133 . . . . . . 7 ((𝑋𝐴𝑌𝐴𝑍𝐴) → 𝑍𝐴)
2 simp2 1132 . . . . . . 7 ((𝑋𝐴𝑌𝐴𝑍𝐴) → 𝑌𝐴)
31, 2jca 555 . . . . . 6 ((𝑋𝐴𝑌𝐴𝑍𝐴) → (𝑍𝐴𝑌𝐴))
4 sotric 5209 . . . . . 6 ((𝑅 Or 𝐴 ∧ (𝑍𝐴𝑌𝐴)) → (𝑍𝑅𝑌 ↔ ¬ (𝑍 = 𝑌𝑌𝑅𝑍)))
53, 4sylan2 492 . . . . 5 ((𝑅 Or 𝐴 ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) → (𝑍𝑅𝑌 ↔ ¬ (𝑍 = 𝑌𝑌𝑅𝑍)))
65con2bid 343 . . . 4 ((𝑅 Or 𝐴 ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) → ((𝑍 = 𝑌𝑌𝑅𝑍) ↔ ¬ 𝑍𝑅𝑌))
76adantr 472 . . 3 (((𝑅 Or 𝐴 ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) ∧ 𝑋𝑅𝑌) → ((𝑍 = 𝑌𝑌𝑅𝑍) ↔ ¬ 𝑍𝑅𝑌))
8 breq2 4804 . . . . . 6 (𝑍 = 𝑌 → (𝑋𝑅𝑍𝑋𝑅𝑌))
98biimprcd 240 . . . . 5 (𝑋𝑅𝑌 → (𝑍 = 𝑌𝑋𝑅𝑍))
109adantl 473 . . . 4 (((𝑅 Or 𝐴 ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) ∧ 𝑋𝑅𝑌) → (𝑍 = 𝑌𝑋𝑅𝑍))
11 sotr 5205 . . . . 5 ((𝑅 Or 𝐴 ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) → ((𝑋𝑅𝑌𝑌𝑅𝑍) → 𝑋𝑅𝑍))
1211expdimp 452 . . . 4 (((𝑅 Or 𝐴 ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) ∧ 𝑋𝑅𝑌) → (𝑌𝑅𝑍𝑋𝑅𝑍))
1310, 12jaod 394 . . 3 (((𝑅 Or 𝐴 ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) ∧ 𝑋𝑅𝑌) → ((𝑍 = 𝑌𝑌𝑅𝑍) → 𝑋𝑅𝑍))
147, 13sylbird 250 . 2 (((𝑅 Or 𝐴 ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) ∧ 𝑋𝑅𝑌) → (¬ 𝑍𝑅𝑌𝑋𝑅𝑍))
1514expimpd 630 1 ((𝑅 Or 𝐴 ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) → ((𝑋𝑅𝑌 ∧ ¬ 𝑍𝑅𝑌) → 𝑋𝑅𝑍))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383  w3a 1072   = wceq 1628  wcel 2135   class class class wbr 4800   Or wor 5182
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ral 3051  df-rab 3055  df-v 3338  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-nul 4055  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-br 4801  df-po 5183  df-so 5184
This theorem is referenced by:  nosupbnd2  32164  sltletr  32183
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