Theorem reneg1lt0

Description: Negative one is a negative number. (Contributed by SN, 1-Jun-2024)

	Ref	Expression
Assertion	reneg1lt0	⊢ ( 0 −ℝ 1 ) < 0

Proof

Step	Hyp	Ref	Expression
1		sn-0lt1	⊢ 0 < 1
2		1re	⊢ 1 ∈ ℝ
3		relt0neg2	⊢ ( 1 ∈ ℝ → ( 0 < 1 ↔ ( 0 −ℝ 1 ) < 0 ) )
4	2 3	ax-mp	⊢ ( 0 < 1 ↔ ( 0 −ℝ 1 ) < 0 )
5	1 4	mpbi	⊢ ( 0 −ℝ 1 ) < 0