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Theorem sotr3 31994
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 1132 . . . . . . 7 ((𝑋𝐴𝑌𝐴𝑍𝐴) → 𝑍𝐴)
2 simp2 1131 . . . . . . 7 ((𝑋𝐴𝑌𝐴𝑍𝐴) → 𝑌𝐴)
31, 2jca 501 . . . . . 6 ((𝑋𝐴𝑌𝐴𝑍𝐴) → (𝑍𝐴𝑌𝐴))
4 sotric 5196 . . . . . 6 ((𝑅 Or 𝐴 ∧ (𝑍𝐴𝑌𝐴)) → (𝑍𝑅𝑌 ↔ ¬ (𝑍 = 𝑌𝑌𝑅𝑍)))
53, 4sylan2 580 . . . . 5 ((𝑅 Or 𝐴 ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) → (𝑍𝑅𝑌 ↔ ¬ (𝑍 = 𝑌𝑌𝑅𝑍)))
65con2bid 343 . . . 4 ((𝑅 Or 𝐴 ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) → ((𝑍 = 𝑌𝑌𝑅𝑍) ↔ ¬ 𝑍𝑅𝑌))
76adantr 466 . . 3 (((𝑅 Or 𝐴 ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) ∧ 𝑋𝑅𝑌) → ((𝑍 = 𝑌𝑌𝑅𝑍) ↔ ¬ 𝑍𝑅𝑌))
8 breq2 4790 . . . . . 6 (𝑍 = 𝑌 → (𝑋𝑅𝑍𝑋𝑅𝑌))
98biimprcd 240 . . . . 5 (𝑋𝑅𝑌 → (𝑍 = 𝑌𝑋𝑅𝑍))
109adantl 467 . . . 4 (((𝑅 Or 𝐴 ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) ∧ 𝑋𝑅𝑌) → (𝑍 = 𝑌𝑋𝑅𝑍))
11 sotr 5192 . . . . 5 ((𝑅 Or 𝐴 ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) → ((𝑋𝑅𝑌𝑌𝑅𝑍) → 𝑋𝑅𝑍))
1211expdimp 440 . . . 4 (((𝑅 Or 𝐴 ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) ∧ 𝑋𝑅𝑌) → (𝑌𝑅𝑍𝑋𝑅𝑍))
1310, 12jaod 848 . . 3 (((𝑅 Or 𝐴 ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) ∧ 𝑋𝑅𝑌) → ((𝑍 = 𝑌𝑌𝑅𝑍) → 𝑋𝑅𝑍))
147, 13sylbird 250 . 2 (((𝑅 Or 𝐴 ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) ∧ 𝑋𝑅𝑌) → (¬ 𝑍𝑅𝑌𝑋𝑅𝑍))
1514expimpd 441 1 ((𝑅 Or 𝐴 ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) → ((𝑋𝑅𝑌 ∧ ¬ 𝑍𝑅𝑌) → 𝑋𝑅𝑍))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  wo 836  w3a 1071   = wceq 1631  wcel 2145   class class class wbr 4786   Or wor 5169
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-po 5170  df-so 5171
This theorem is referenced by:  nosupbnd2  32199  sltletr  32218
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