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Theorem sotrd 34377
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 5574 . . 3 ((𝑅 Or 𝐴 ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) → ((𝑋𝑅𝑌𝑌𝑅𝑍) → 𝑋𝑅𝑍))
83, 4, 5, 6, 7syl13anc 1373 . 2 (𝜑 → ((𝑋𝑅𝑌𝑌𝑅𝑍) → 𝑋𝑅𝑍))
91, 2, 8mp2and 698 1 (𝜑𝑋𝑅𝑍)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wcel 2107   class class class wbr 5110   Or wor 5549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-po 5550  df-so 5551
This theorem is referenced by: (None)
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