Theorem sopo 4458

Description: A strict linear order is a strict partial order. (Contributed by NM, 28-Mar-1997.)

Assertion

Ref	Expression
sopo	⊢ (𝑅 Or 𝐴 → 𝑅 Po 𝐴)

Proof of Theorem sopo

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-iso 4442	. 2 ⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦))))
2	1	simplbi 274	1 ⊢ (𝑅 Or 𝐴 → 𝑅 Po 𝐴)

This proof depends on syntax axioms:   → wi 4   ∨ wo 720  ∀wral 2528   class class class wbr 4130   Po wpo 4439   Or wor 4440

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106

This proof depends on definitions:  df-bi 117  df-iso 4442

This theorem is used by:  sonr  4462  sotr  4463  so2nr  4466  so3nr  4467  sosng  4848  fimaxq  11270