Theorem xlt0neg2

Description: Extended real version of lt0neg2 . (Contributed by Mario Carneiro, 20-Aug-2015)

	Ref	Expression
Assertion	xlt0neg2	⊢ ( 𝐴 ∈ ℝ* → ( 0 < 𝐴 ↔ -𝑒 𝐴 < 0 ) )

Proof

Step	Hyp	Ref	Expression
1		0xr	⊢ 0 ∈ ℝ*
2		xltneg	⊢ ( ( 0 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ) → ( 0 < 𝐴 ↔ -𝑒 𝐴 < -𝑒 0 ) )
3	1 2	mpan	⊢ ( 𝐴 ∈ ℝ* → ( 0 < 𝐴 ↔ -𝑒 𝐴 < -𝑒 0 ) )
4		xneg0	⊢ -𝑒 0 = 0
5	4	breq2i	⊢ ( -𝑒 𝐴 < -𝑒 0 ↔ -𝑒 𝐴 < 0 )
6	3 5	syl6bb	⊢ ( 𝐴 ∈ ℝ* → ( 0 < 𝐴 ↔ -𝑒 𝐴 < 0 ) )