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Metamath Proof Explorer

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 )