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	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to A\le if\left(A\le B,B,A\right)$$

Proof

Step	Hyp	Ref	Expression
1		rexr	$$⊢A\in ℝ\to A\in {ℝ}^{*}$$
2		rexr	$$⊢B\in ℝ\to B\in {ℝ}^{*}$$
3		xrmax1	$$⊢\left(A\in {ℝ}^{*}\wedge B\in {ℝ}^{*}\right)\to A\le if\left(A\le B,B,A\right)$$
4	1 2 3	syl2an	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to A\le if\left(A\le B,B,A\right)$$