Theorem max1d

Description: A number is less than or equal to the maximum of it and another. (Contributed by Glauco Siliprandi, 2-Jan-2022)

	Ref	Expression
Hypotheses	max1d.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	max1d.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
Assertion	max1d	⊢ ( 𝜑 → 𝐴 ≤ if ( 𝐴 ≤ 𝐵 , 𝐵 , 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		max1d.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		max1d.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		max1	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → 𝐴 ≤ if ( 𝐴 ≤ 𝐵 , 𝐵 , 𝐴 ) )
4	1 2 3	syl2anc	⊢ ( 𝜑 → 𝐴 ≤ if ( 𝐴 ≤ 𝐵 , 𝐵 , 𝐴 ) )