Theorem max1

Description: A number is less than or equal to the maximum of it and another. See also max1ALT . (Contributed by NM, 3-Apr-2005)

	Ref	Expression
Assertion	max1	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → 𝐴 ≤ if ( 𝐴 ≤ 𝐵 , 𝐵 , 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		rexr	⊢ ( 𝐴 ∈ ℝ → 𝐴 ∈ ℝ* )
2		rexr	⊢ ( 𝐵 ∈ ℝ → 𝐵 ∈ ℝ* )
3		xrmax1	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → 𝐴 ≤ if ( 𝐴 ≤ 𝐵 , 𝐵 , 𝐴 ) )
4	1 2 3	syl2an	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → 𝐴 ≤ if ( 𝐴 ≤ 𝐵 , 𝐵 , 𝐴 ) )