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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