Theorem hashne0

Description: Deduce that the size of a set is not zero. (Contributed by Thierry Arnoux, 26-Oct-2025)

	Ref	Expression
Hypotheses	hashne0.1	$$⊢\phi \to A\in V$$
	hashne0.2	$$⊢\phi \to A\ne \varnothing$$
Assertion	hashne0	$$⊢\phi \to 0<\left|A\right|$$

Proof

Step	Hyp	Ref	Expression
1		hashne0.1	$$⊢\phi \to A\in V$$
2		hashne0.2	$$⊢\phi \to A\ne \varnothing$$
3		hashxnn0	$$⊢A\in V\to \left|A\right|\in {\mathbb{N}}_0^{*}$$
4	1 3	syl	$$⊢\phi \to \left|A\right|\in {\mathbb{N}}_0^{*}$$
5		hasheq0	$$⊢A\in V\to \left(\left|A\right|=0\leftrightarrow A=\varnothing \right)$$
6	5	necon3bid	$$⊢A\in V\to \left(\left|A\right|\ne 0\leftrightarrow A\ne \varnothing \right)$$
7	6	biimpar	$$⊢\left(A\in V\wedge A\ne \varnothing \right)\to \left|A\right|\ne 0$$
8	1 2 7	syl2anc	$$⊢\phi \to \left|A\right|\ne 0$$
9		xnn0gt0	$$⊢\left(\left|A\right|\in {\mathbb{N}}_0^{*}\wedge \left|A\right|\ne 0\right)\to 0<\left|A\right|$$
10	4 8 9	syl2anc	$$⊢\phi \to 0<\left|A\right|$$