Theorem gt-lt

Description: Simple relationship between < and > . (Contributed by David A. Wheeler, 19-Apr-2015) (New usage is discouraged.)

	Ref	Expression
Assertion	gt-lt	$$⊢\left(A\in \mathrm{V}\wedge B\in \mathrm{V}\right)\to \left(A>B\leftrightarrow B<A\right)$$

Proof

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