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Metamath Proof Explorer

Theorem lenegcon2d

Description: Contraposition of negative in 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016)

Ref Expression
Hypotheses leidd.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
ltnegd.2 ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
lenegcon2d.3 ⊢ ( 𝜑 → 𝐴 ≤ - 𝐵 )
Assertion lenegcon2d ( 𝜑 → 𝐵 ≤ - 𝐴 )

Proof

Step Hyp Ref Expression
1 leidd.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2 ltnegd.2 ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3 lenegcon2d.3 ⊢ ( 𝜑 → 𝐴 ≤ - 𝐵 )
4 lenegcon2 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 ≤ - 𝐵 ↔ 𝐵 ≤ - 𝐴 ) )
5 1 2 4 syl2anc ⊢ ( 𝜑 → ( 𝐴 ≤ - 𝐵 ↔ 𝐵 ≤ - 𝐴 ) )
6 3 5 mpbid ⊢ ( 𝜑 → 𝐵 ≤ - 𝐴 )