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Theorem sotrd 33184
 Description: Transitivity law for strict orderings, deduction form. (Contributed by Scott Fenton, 24-Nov-2021.)
Hypotheses
Ref Expression
sotrd.1 (𝜑𝑅 Or 𝐴)
sotrd.2 (𝜑𝑋𝐴)
sotrd.3 (𝜑𝑌𝐴)
sotrd.4 (𝜑𝑍𝐴)
sotrd.5 (𝜑𝑋𝑅𝑌)
sotrd.6 (𝜑𝑌𝑅𝑍)
Assertion
Ref Expression
sotrd (𝜑𝑋𝑅𝑍)

Proof of Theorem sotrd
StepHypRef Expression
1 sotrd.5 . 2 (𝜑𝑋𝑅𝑌)
2 sotrd.6 . 2 (𝜑𝑌𝑅𝑍)
3 sotrd.1 . . 3 (𝜑𝑅 Or 𝐴)
4 sotrd.2 . . 3 (𝜑𝑋𝐴)
5 sotrd.3 . . 3 (𝜑𝑌𝐴)
6 sotrd.4 . . 3 (𝜑𝑍𝐴)
7 sotr 5465 . . 3 ((𝑅 Or 𝐴 ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) → ((𝑋𝑅𝑌𝑌𝑅𝑍) → 𝑋𝑅𝑍))
83, 4, 5, 6, 7syl13anc 1369 . 2 (𝜑 → ((𝑋𝑅𝑌𝑌𝑅𝑍) → 𝑋𝑅𝑍))
91, 2, 8mp2and 698 1 (𝜑𝑋𝑅𝑍)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2111   class class class wbr 5034   Or wor 5441 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 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-v 3444  df-un 3888  df-sn 4529  df-pr 4531  df-op 4535  df-br 5035  df-po 5442  df-so 5443 This theorem is referenced by: (None)
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