Theorem ltled

Description: 'Less than' implies 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	ltd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	ltd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
	ltled.1	⊢ ( 𝜑 → 𝐴 < 𝐵 )
Assertion	ltled	⊢ ( 𝜑 → 𝐴 ≤ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		ltd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		ltd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		ltled.1	⊢ ( 𝜑 → 𝐴 < 𝐵 )
4		ltle	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 < 𝐵 → 𝐴 ≤ 𝐵 ) )
5	1 2 4	syl2anc	⊢ ( 𝜑 → ( 𝐴 < 𝐵 → 𝐴 ≤ 𝐵 ) )
6	3 5	mpd	⊢ ( 𝜑 → 𝐴 ≤ 𝐵 )