Theorem gt-lth

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

	Ref	Expression
Hypotheses	gt-lth.1	⊢ 𝐴 ∈ V
	gt-lth.2	⊢ 𝐵 ∈ V
Assertion	gt-lth	⊢ ( 𝐴 > 𝐵 ↔ 𝐵 < 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		gt-lth.1	⊢ 𝐴 ∈ V
2		gt-lth.2	⊢ 𝐵 ∈ V
3		df-gt	⊢ > = ◡ <
4	3	breqi	⊢ ( 𝐴 > 𝐵 ↔ 𝐴 ◡ < 𝐵 )
5	1 2	brcnv	⊢ ( 𝐴 ◡ < 𝐵 ↔ 𝐵 < 𝐴 )
6	4 5	bitri	⊢ ( 𝐴 > 𝐵 ↔ 𝐵 < 𝐴 )