Theorem sotrd 34723

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 5611	. . 3 ⊢ ((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴)) → ((𝑋𝑅𝑌 ∧ 𝑌𝑅𝑍) → 𝑋𝑅𝑍))
8	3, 4, 5, 6, 7	syl13anc 1372	. 2 ⊢ (𝜑 → ((𝑋𝑅𝑌 ∧ 𝑌𝑅𝑍) → 𝑋𝑅𝑍))
9	1, 2, 8	mp2and 697	1 ⊢ (𝜑 → 𝑋𝑅𝑍)

Syntax hints:   → wi 4   ∧ wa 396   ∈ wcel 2106   class class class wbr 5147   Or wor 5586

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-po 5587  df-so 5588

This theorem is referenced by: (None)