Theorem sotrd 5617

Description: Transitivity law for strict orderings, deduction form. (Contributed by Scott Fenton, 24-Nov-2021.)

Hypotheses

Ref	Expression
sotrd.1	⊢ (𝜑 → 𝑅 Or 𝐴)
sotrd.2	⊢ (𝜑 → 𝑋 ∈ 𝐴)
sotrd.3	⊢ (𝜑 → 𝑌 ∈ 𝐴)
sotrd.4	⊢ (𝜑 → 𝑍 ∈ 𝐴)
sotrd.5	⊢ (𝜑 → 𝑋𝑅𝑌)
sotrd.6	⊢ (𝜑 → 𝑌𝑅𝑍)

Assertion

Ref	Expression
sotrd	⊢ (𝜑 → 𝑋𝑅𝑍)

Proof of Theorem sotrd

Step	Hyp	Ref	Expression
1		sotrd.5	. 2 ⊢ (𝜑 → 𝑋𝑅𝑌)
2		sotrd.6	. 2 ⊢ (𝜑 → 𝑌𝑅𝑍)
3		sotrd.1	. . 3 ⊢ (𝜑 → 𝑅 Or 𝐴)
4		sotrd.2	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
5		sotrd.3	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐴)
6		sotrd.4	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐴)
7		sotr 5616	. . 3 ⊢ ((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴)) → ((𝑋𝑅𝑌 ∧ 𝑌𝑅𝑍) → 𝑋𝑅𝑍))
8	3, 4, 5, 6, 7	syl13anc 1373	. 2 ⊢ (𝜑 → ((𝑋𝑅𝑌 ∧ 𝑌𝑅𝑍) → 𝑋𝑅𝑍))
9	1, 2, 8	mp2and 699	1 ⊢ (𝜑 → 𝑋𝑅𝑍)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2107   class class class wbr 5142   Or wor 5590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-po 5591  df-so 5592

This theorem is referenced by:  ormkglobd  46895