Theorem min1

Description: The minimum of two numbers is less than or equal to the first. (Contributed by NM, 3-Aug-2007)

	Ref	Expression
Assertion	min1	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to if\left(A\le B,A,B\right)\le A$$

Proof

Step	Hyp	Ref	Expression
1		rexr	$$⊢A\in ℝ\to A\in {ℝ}^{*}$$
2		rexr	$$⊢B\in ℝ\to B\in {ℝ}^{*}$$
3		xrmin1	$$⊢\left(A\in {ℝ}^{*}\wedge B\in {ℝ}^{*}\right)\to if\left(A\le B,A,B\right)\le A$$
4	1 2 3	syl2an	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to if\left(A\le B,A,B\right)\le A$$