Theorem leltne

Description: 'Less than or equal to' implies 'less than' is not 'equals'. (Contributed by NM, 27-Jul-1999)

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

Proof

Step	Hyp	Ref	Expression
1		lttri3	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 = 𝐵 ↔ ( ¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴 ) ) )
2		simpl	⊢ ( ( ¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴 ) → ¬ 𝐴 < 𝐵 )
3	1 2	biimtrdi	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 = 𝐵 → ¬ 𝐴 < 𝐵 ) )
4	3	adantr	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ 𝐴 ≤ 𝐵 ) → ( 𝐴 = 𝐵 → ¬ 𝐴 < 𝐵 ) )
5		leloe	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 ≤ 𝐵 ↔ ( 𝐴 < 𝐵 ∨ 𝐴 = 𝐵 ) ) )
6	5	biimpa	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ 𝐴 ≤ 𝐵 ) → ( 𝐴 < 𝐵 ∨ 𝐴 = 𝐵 ) )
7	6	ord	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ 𝐴 ≤ 𝐵 ) → ( ¬ 𝐴 < 𝐵 → 𝐴 = 𝐵 ) )
8	4 7	impbid	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ 𝐴 ≤ 𝐵 ) → ( 𝐴 = 𝐵 ↔ ¬ 𝐴 < 𝐵 ) )
9	8	necon2abid	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ 𝐴 ≤ 𝐵 ) → ( 𝐴 < 𝐵 ↔ 𝐴 ≠ 𝐵 ) )
10		necom	⊢ ( 𝐵 ≠ 𝐴 ↔ 𝐴 ≠ 𝐵 )
11	9 10	bitr4di	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ 𝐴 ≤ 𝐵 ) → ( 𝐴 < 𝐵 ↔ 𝐵 ≠ 𝐴 ) )
12	11	3impa	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ) → ( 𝐴 < 𝐵 ↔ 𝐵 ≠ 𝐴 ) )