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Metamath Proof Explorer

Theorem ltnrd

Description: 'Less than' is irreflexive. (Contributed by Mario Carneiro, 27-May-2016)

Ref Expression
Hypothesis ltd.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
Assertion ltnrd ( 𝜑 → ¬ 𝐴 < 𝐴 )

Proof

Step Hyp Ref Expression
1 ltd.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2 ltnr ⊢ ( 𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴 )
3 1 2 syl ⊢ ( 𝜑 → ¬ 𝐴 < 𝐴 )