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Metamath Proof Explorer

Theorem ltnsymd

Description: 'Less than' implies 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016)

Ref Expression
Hypotheses ltd.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
ltd.2 ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
ltled.1 ⊢ ( 𝜑 → 𝐴 < 𝐵 )
Assertion ltnsymd ( 𝜑 → ¬ 𝐵 < 𝐴 )

Proof

Step Hyp Ref Expression
1 ltd.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2 ltd.2 ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3 ltled.1 ⊢ ( 𝜑 → 𝐴 < 𝐵 )
4 1 2 3 ltled ⊢ ( 𝜑 → 𝐴 ≤ 𝐵 )
5 1 2 lenltd ⊢ ( 𝜑 → ( 𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴 ) )
6 4 5 mpbid ⊢ ( 𝜑 → ¬ 𝐵 < 𝐴 )