/sci/ - Science & Math

I need to show that [math]|x|<1[/math] iff [math]-1<x<1[/math].

PROOF

[math]|x|<1[/math]
iff [math]|x|<1 \wedge \top[/math]
iff [math]|x|<1 \wedge (x<0 \veebar x\geq 0)[/math]
iff [math](|x|<1 \wedge x<0)\veebar (|x|<1 \wedge x\geq 0)[/math]
iff [math](-x<1 \wedge x<0) \veebar (x<1 \wedge x\geq 0)[/math]
iff [math](-1<x \wedge x<0) \veebar (x<1 \wedge x\geq 0)[/math]
iff [math]-1<x<0 \veebar 0\leq x <1 [/math]
iff ...
iff [math]-1<x<1[/math].

How about the last step?
([math]-1<x<1[/math] means [math]-1<x \vee x <1[/math])