Theorem gte-lteh

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

	Ref	Expression
Hypotheses	gte-lteh.1	⊢ 𝐴 ∈ V
	gte-lteh.2	⊢ 𝐵 ∈ V
Assertion	gte-lteh	⊢ ( 𝐴 ≥ 𝐵 ↔ 𝐵 ≤ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		gte-lteh.1	⊢ 𝐴 ∈ V
2		gte-lteh.2	⊢ 𝐵 ∈ V
3		df-gte	⊢ ≥ = ◡ ≤
4	3	breqi	⊢ ( 𝐴 ≥ 𝐵 ↔ 𝐴 ◡ ≤ 𝐵 )
5	1 2	brcnv	⊢ ( 𝐴 ◡ ≤ 𝐵 ↔ 𝐵 ≤ 𝐴 )
6	4 5	bitri	⊢ ( 𝐴 ≥ 𝐵 ↔ 𝐵 ≤ 𝐴 )