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	⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( 𝐴 > 𝐵 ↔ 𝐵 < 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		df-gt	⊢ > = ◡ <
2	1	breqi	⊢ ( 𝐴 > 𝐵 ↔ 𝐴 ◡ < 𝐵 )
3		brcnvg	⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( 𝐴 ◡ < 𝐵 ↔ 𝐵 < 𝐴 ) )
4	2 3	syl5bb	⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( 𝐴 > 𝐵 ↔ 𝐵 < 𝐴 ) )