Theorem xrgtned

Description: 'Greater than' implies not equal. (Contributed by Glauco Siliprandi, 17-Aug-2020)

	Ref	Expression
Hypotheses	xrgtned.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ* )
	xrgtned.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ* )
	xrgtned.3	⊢ ( 𝜑 → 𝐴 < 𝐵 )
Assertion	xrgtned	⊢ ( 𝜑 → 𝐵 ≠ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		xrgtned.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ* )
2		xrgtned.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ* )
3		xrgtned.3	⊢ ( 𝜑 → 𝐴 < 𝐵 )
4		xrltne	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵 ) → 𝐵 ≠ 𝐴 )
5	1 2 3 4	syl3anc	⊢ ( 𝜑 → 𝐵 ≠ 𝐴 )