Theorem sotrd 5592

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 5591	. . 3 ⊢ ((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴)) → ((𝑋𝑅𝑌 ∧ 𝑌𝑅𝑍) → 𝑋𝑅𝑍))
8	3, 4, 5, 6, 7	syl13anc 1374	. 2 ⊢ (𝜑 → ((𝑋𝑅𝑌 ∧ 𝑌𝑅𝑍) → 𝑋𝑅𝑍))
9	1, 2, 8	mp2and 699	1 ⊢ (𝜑 → 𝑋𝑅𝑍)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2109   class class class wbr 5124   Or wor 5565

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5125  df-po 5566  df-so 5567

This theorem is referenced by:  ormkglobd  46871