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	⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( 𝐴 ≥ 𝐵 ↔ 𝐵 ≤ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		df-gte	⊢ ≥ = ◡ ≤
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 ) )