Theorem hashelne0d

Description: A set with an element has nonzero size. (Contributed by Rohan Ridenour, 3-Aug-2023)

	Ref	Expression
Hypotheses	hashelne0d.1	$$⊢\phi \to B\in A$$
	hashelne0d.2	$$⊢\phi \to A\in V$$
Assertion	hashelne0d	$$⊢\phi \to \neg \left|A\right|=0$$

Proof

Step	Hyp	Ref	Expression
1		hashelne0d.1	$$⊢\phi \to B\in A$$
2		hashelne0d.2	$$⊢\phi \to A\in V$$
3	1	ne0d	$$⊢\phi \to A\ne \varnothing$$
4	3	neneqd	$$⊢\phi \to \neg A=\varnothing$$
5		hasheq0	$$⊢A\in V\to \left(\left|A\right|=0\leftrightarrow A=\varnothing \right)$$
6	2 5	syl	$$⊢\phi \to \left(\left|A\right|=0\leftrightarrow A=\varnothing \right)$$
7	4 6	mtbird	$$⊢\phi \to \neg \left|A\right|=0$$