Theorem lenlteq

Description: 'less than or equal to' but not 'less than' implies 'equal' . (Contributed by Glauco Siliprandi, 3-Mar-2021)

	Ref	Expression
Hypotheses	lenlteq.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	lenlteq.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
	lenlteq.3	⊢ ( 𝜑 → 𝐴 ≤ 𝐵 )
	lenlteq.4	⊢ ( 𝜑 → ¬ 𝐴 < 𝐵 )
Assertion	lenlteq	⊢ ( 𝜑 → 𝐴 = 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		lenlteq.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		lenlteq.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		lenlteq.3	⊢ ( 𝜑 → 𝐴 ≤ 𝐵 )
4		lenlteq.4	⊢ ( 𝜑 → ¬ 𝐴 < 𝐵 )
5	3 4	jca	⊢ ( 𝜑 → ( 𝐴 ≤ 𝐵 ∧ ¬ 𝐴 < 𝐵 ) )
6		eqlelt	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 = 𝐵 ↔ ( 𝐴 ≤ 𝐵 ∧ ¬ 𝐴 < 𝐵 ) ) )
7	1 2 6	syl2anc	⊢ ( 𝜑 → ( 𝐴 = 𝐵 ↔ ( 𝐴 ≤ 𝐵 ∧ ¬ 𝐴 < 𝐵 ) ) )
8	5 7	mpbird	⊢ ( 𝜑 → 𝐴 = 𝐵 )