Theorem absord

Description: The absolute value of a real number is either that number or its negative. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypothesis	resqrcld.1	$$⊢\phi \to A\in ℝ$$
Assertion	absord	$$⊢\phi \to \left(\left|A\right|=A\vee \left|A\right|=-A\right)$$

Proof

Step	Hyp	Ref	Expression
1		resqrcld.1	$$⊢\phi \to A\in ℝ$$
2		absor	$$⊢A\in ℝ\to \left(\left|A\right|=A\vee \left|A\right|=-A\right)$$
3	1 2	syl	$$⊢\phi \to \left(\left|A\right|=A\vee \left|A\right|=-A\right)$$