Theorem xrltned

Description: 'Less than' implies not equal. (Contributed by Glauco Siliprandi, 21-Nov-2020)

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

Proof

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