Theorem gte-lte

Description: Simple relationship between <_ and >_ . (Contributed by David A. Wheeler, 10-May-2015) (New usage is discouraged.)

	Ref	Expression
Assertion	gte-lte	$$⊢\left(A\in \mathrm{V}\wedge B\in \mathrm{V}\right)\to \left(A\ge B\leftrightarrow B\le A\right)$$

Proof

Step	Hyp	Ref	Expression
1		df-gte	$$⊢\ge ={\le }^{-1}$$
2	1	breqi	$$⊢A\ge B\leftrightarrow A{\le }^{-1}B$$
3		brcnvg	$$⊢\left(A\in \mathrm{V}\wedge B\in \mathrm{V}\right)\to \left(A{\le }^{-1}B\leftrightarrow B\le A\right)$$
4	2 3	syl5bb	$$⊢\left(A\in \mathrm{V}\wedge B\in \mathrm{V}\right)\to \left(A\ge B\leftrightarrow B\le A\right)$$