Theorem gtned

Description: 'Less than' implies not equal. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	ltd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	ltned.2	⊢ ( 𝜑 → 𝐴 < 𝐵 )
Assertion	gtned	⊢ ( 𝜑 → 𝐵 ≠ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		ltd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		ltned.2	⊢ ( 𝜑 → 𝐴 < 𝐵 )
3		ltne	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐴 < 𝐵 ) → 𝐵 ≠ 𝐴 )
4	1 2 3	syl2anc	⊢ ( 𝜑 → 𝐵 ≠ 𝐴 )