Theorem issoi

Description: An irreflexive, transitive, linear relation is a strict ordering. (Contributed by NM, 21-Jan-1996) (Revised by Mario Carneiro, 9-Jul-2014)

	Ref	Expression
Hypotheses	issoi.1	⊢ ( 𝑥 ∈ 𝐴 → ¬ 𝑥 𝑅 𝑥 )
	issoi.2	⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) → ( ( 𝑥 𝑅 𝑦 ∧ 𝑦 𝑅 𝑧 ) → 𝑥 𝑅 𝑧 ) )
	issoi.3	⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) → ( 𝑥 𝑅 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 𝑅 𝑥 ) )
Assertion	issoi	⊢ 𝑅 Or 𝐴

Proof

Step	Hyp	Ref	Expression
1		issoi.1	⊢ ( 𝑥 ∈ 𝐴 → ¬ 𝑥 𝑅 𝑥 )
2		issoi.2	⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) → ( ( 𝑥 𝑅 𝑦 ∧ 𝑦 𝑅 𝑧 ) → 𝑥 𝑅 𝑧 ) )
3		issoi.3	⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) → ( 𝑥 𝑅 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 𝑅 𝑥 ) )
4	1	adantl	⊢ ( ( ⊤ ∧ 𝑥 ∈ 𝐴 ) → ¬ 𝑥 𝑅 𝑥 )
5	2	adantl	⊢ ( ( ⊤ ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ) → ( ( 𝑥 𝑅 𝑦 ∧ 𝑦 𝑅 𝑧 ) → 𝑥 𝑅 𝑧 ) )
6	4 5	ispod	⊢ ( ⊤ → 𝑅 Po 𝐴 )
7	3	adantl	⊢ ( ( ⊤ ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) → ( 𝑥 𝑅 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 𝑅 𝑥 ) )
8	6 7	issod	⊢ ( ⊤ → 𝑅 Or 𝐴 )
9	8	mptru	⊢ 𝑅 Or 𝐴