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Metamath Proof Explorer

Theorem ltnr

Description: 'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999)

		Ref	Expression
	Assertion	ltnr	⊢ ( 𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴 )

Step	Hyp	Ref	Expression
1		ltso	⊢ < Or ℝ
2		sonr	⊢ ( ( < Or ℝ ∧ 𝐴 ∈ ℝ ) → ¬ 𝐴 < 𝐴 )
3	1 2	mpan	⊢ ( 𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴 )