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Metamath Proof Explorer

Theorem ltlen

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

Ref Expression
Assertion ltlen ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 < 𝐵 ↔ ( 𝐴 ≤ 𝐵 ∧ 𝐵 ≠ 𝐴 ) ) )

Proof

Step Hyp Ref Expression
1 ltle ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 < 𝐵 → 𝐴 ≤ 𝐵 ) )
2 ltne ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐴 < 𝐵 ) → 𝐵 ≠ 𝐴 )
3 2 ex ⊢ ( 𝐴 ∈ ℝ → ( 𝐴 < 𝐵 → 𝐵 ≠ 𝐴 ) )
4 3 adantr ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 < 𝐵 → 𝐵 ≠ 𝐴 ) )
5 1 4 jcad ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 < 𝐵 → ( 𝐴 ≤ 𝐵 ∧ 𝐵 ≠ 𝐴 ) ) )
6 leloe ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 ≤ 𝐵 ↔ ( 𝐴 < 𝐵 ∨ 𝐴 = 𝐵 ) ) )
7 df-ne ⊢ ( 𝐵 ≠ 𝐴 ↔ ¬ 𝐵 = 𝐴 )
8 pm2.24 ⊢ ( 𝐵 = 𝐴 → ( ¬ 𝐵 = 𝐴 → 𝐴 < 𝐵 ) )
9 8 eqcoms ⊢ ( 𝐴 = 𝐵 → ( ¬ 𝐵 = 𝐴 → 𝐴 < 𝐵 ) )
10 7 9 biimtrid ⊢ ( 𝐴 = 𝐵 → ( 𝐵 ≠ 𝐴 → 𝐴 < 𝐵 ) )
11 10 jao1i ⊢ ( ( 𝐴 < 𝐵 ∨ 𝐴 = 𝐵 ) → ( 𝐵 ≠ 𝐴 → 𝐴 < 𝐵 ) )
12 6 11 biimtrdi ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 ≤ 𝐵 → ( 𝐵 ≠ 𝐴 → 𝐴 < 𝐵 ) ) )
13 12 impd ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( 𝐴 ≤ 𝐵 ∧ 𝐵 ≠ 𝐴 ) → 𝐴 < 𝐵 ) )
14 5 13 impbid ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 < 𝐵 ↔ ( 𝐴 ≤ 𝐵 ∧ 𝐵 ≠ 𝐴 ) ) )