Theorem abs00i

Description: The absolute value of a number is zero iff the number is zero. Proposition 10-3.7(c) of Gleason p. 133. (Contributed by NM, 28-Jul-1999)

	Ref	Expression
Hypothesis	absvalsqi.1	$$⊢A\in ℂ$$
Assertion	abs00i	$$⊢\left|A\right|=0\leftrightarrow A=0$$

Proof

Step	Hyp	Ref	Expression
1		absvalsqi.1	$$⊢A\in ℂ$$
2		abs00	$$⊢A\in ℂ\to \left(\left|A\right|=0\leftrightarrow A=0\right)$$
3	1 2	ax-mp	$$⊢\left|A\right|=0\leftrightarrow A=0$$