Theorem absne0d

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 Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypotheses	abscld.1	$$⊢\phi \to A\in ℂ$$
	absne0d.2	$$⊢\phi \to A\ne 0$$
Assertion	absne0d	$$⊢\phi \to \left|A\right|\ne 0$$

Proof

Step	Hyp	Ref	Expression
1		abscld.1	$$⊢\phi \to A\in ℂ$$
2		absne0d.2	$$⊢\phi \to A\ne 0$$
3	1	abs00ad	$$⊢\phi \to \left(\left|A\right|=0\leftrightarrow A=0\right)$$
4	3	necon3bid	$$⊢\phi \to \left(\left|A\right|\ne 0\leftrightarrow A\ne 0\right)$$
5	2 4	mpbird	$$⊢\phi \to \left|A\right|\ne 0$$