Theorem sotrd 33732

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 5527	. . 3 ⊢ ((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴)) → ((𝑋𝑅𝑌 ∧ 𝑌𝑅𝑍) → 𝑋𝑅𝑍))
8	3, 4, 5, 6, 7	syl13anc 1371	. 2 ⊢ (𝜑 → ((𝑋𝑅𝑌 ∧ 𝑌𝑅𝑍) → 𝑋𝑅𝑍))
9	1, 2, 8	mp2and 696	1 ⊢ (𝜑 → 𝑋𝑅𝑍)

Syntax hints:   → wi 4   ∧ wa 396   ∈ wcel 2106   class class class wbr 5074   Or wor 5502

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-po 5503  df-so 5504

This theorem is referenced by: (None)