Theorem hash1n0

Description: If the size of a set is 1 the set is not empty. (Contributed by AV, 23-Dec-2020)

	Ref	Expression
Assertion	hash1n0	$$⊢\left(A\in V\wedge \left|A\right|=1\right)\to A\ne \varnothing$$

Proof

Step	Hyp	Ref	Expression
1		hash1snb	$$⊢A\in V\to \left(\left|A\right|=1\leftrightarrow \exists aA=\left\{a\right\}\right)$$
2		id	$$⊢A=\left\{a\right\}\to A=\left\{a\right\}$$
3		vex	$$⊢a\in \mathrm{V}$$
4	3	snnz	$$⊢\left\{a\right\}\ne \varnothing$$
5	4	a1i	$$⊢A=\left\{a\right\}\to \left\{a\right\}\ne \varnothing$$
6	2 5	eqnetrd	$$⊢A=\left\{a\right\}\to A\ne \varnothing$$
7	6	exlimiv	$$⊢\exists aA=\left\{a\right\}\to A\ne \varnothing$$
8	1 7	biimtrdi	$$⊢A\in V\to \left(\left|A\right|=1\to A\ne \varnothing \right)$$
9	8	imp	$$⊢\left(A\in V\wedge \left|A\right|=1\right)\to A\ne \varnothing$$