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Metamath Proof Explorer

Theorem ltleni

Description: 'Less than' expressed in terms of 'less than or equal to'. (Contributed by NM, 27-Oct-1999)

Ref Expression
Hypotheses lt.1 ⊢ 𝐴 ∈ ℝ
lt.2 ⊢ 𝐵 ∈ ℝ
Assertion ltleni ( 𝐴 < 𝐵 ↔ ( 𝐴 ≤ 𝐵 ∧ 𝐵 ≠ 𝐴 ) )

Proof

Step Hyp Ref Expression
1 lt.1 ⊢ 𝐴 ∈ ℝ
2 lt.2 ⊢ 𝐵 ∈ ℝ
3 ltlen ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 < 𝐵 ↔ ( 𝐴 ≤ 𝐵 ∧ 𝐵 ≠ 𝐴 ) ) )
4 1 2 3 mp2an ⊢ ( 𝐴 < 𝐵 ↔ ( 𝐴 ≤ 𝐵 ∧ 𝐵 ≠ 𝐴 ) )