Theorem max2d

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	max2d.1	$$⊢\phi \to A\in ℝ$$
	max2d.2	$$⊢\phi \to B\in ℝ$$
Assertion	max2d	$$⊢\phi \to B\le if\left(A\le B,B,A\right)$$

Proof

Step	Hyp	Ref	Expression
1		max2d.1	$$⊢\phi \to A\in ℝ$$
2		max2d.2	$$⊢\phi \to B\in ℝ$$
3		max2	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to B\le if\left(A\le B,B,A\right)$$
4	1 2 3	syl2anc	$$⊢\phi \to B\le if\left(A\le B,B,A\right)$$