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	$$⊢\phi \to A\in ℝ$$
	max1d.2	$$⊢\phi \to B\in ℝ$$
Assertion	max1d	$$⊢\phi \to A\le if\left(A\le B,B,A\right)$$

Proof

Step	Hyp	Ref	Expression
1		max1d.1	$$⊢\phi \to A\in ℝ$$
2		max1d.2	$$⊢\phi \to B\in ℝ$$
3		max1	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to A\le if\left(A\le B,B,A\right)$$
4	1 2 3	syl2anc	$$⊢\phi \to A\le if\left(A\le B,B,A\right)$$