Theorem max1ALT

Description: A number is less than or equal to the maximum of it and another. This version of max1 omits the B e. RR antecedent. Although it doesn't exploit undefined behavior, it is still considered poor style, and the use of max1 is preferred. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by NM, 3-Apr-2005)

	Ref	Expression
Assertion	max1ALT	$$⊢A\in ℝ\to A\le if\left(A\le B,B,A\right)$$

Proof

Step	Hyp	Ref	Expression
1		leid	$$⊢A\in ℝ\to A\le A$$
2		iffalse	$$⊢\neg A\le B\to if\left(A\le B,B,A\right)=A$$
3	2	breq2d	$$⊢\neg A\le B\to \left(A\le if\left(A\le B,B,A\right)\leftrightarrow A\le A\right)$$
4	1 3	syl5ibrcom	$$⊢A\in ℝ\to \left(\neg A\le B\to A\le if\left(A\le B,B,A\right)\right)$$
5		id	$$⊢A\le B\to A\le B$$
6		iftrue	$$⊢A\le B\to if\left(A\le B,B,A\right)=B$$
7	5 6	breqtrrd	$$⊢A\le B\to A\le if\left(A\le B,B,A\right)$$
8	4 7	pm2.61d2	$$⊢A\in ℝ\to A\le if\left(A\le B,B,A\right)$$