Theorem hashge1

Description: The cardinality of a nonempty set is greater than or equal to one. (Contributed by Thierry Arnoux, 20-Jun-2017)

	Ref	Expression
Assertion	hashge1	$$⊢\left(A\in V\wedge A\ne \varnothing \right)\to 1\le \left|A\right|$$

Proof

Step	Hyp	Ref	Expression
1		simpr	$$⊢\left(\left(A\in V\wedge A\ne \varnothing \right)\wedge A\in \mathrm{Fin}\right)\to A\in \mathrm{Fin}$$
2		simplr	$$⊢\left(\left(A\in V\wedge A\ne \varnothing \right)\wedge A\in \mathrm{Fin}\right)\to A\ne \varnothing$$
3		hashnncl	$$⊢A\in \mathrm{Fin}\to \left(\left|A\right|\in \mathbb{N}\leftrightarrow A\ne \varnothing \right)$$
4	3	biimpar	$$⊢\left(A\in \mathrm{Fin}\wedge A\ne \varnothing \right)\to \left|A\right|\in \mathbb{N}$$
5	1 2 4	syl2anc	$$⊢\left(\left(A\in V\wedge A\ne \varnothing \right)\wedge A\in \mathrm{Fin}\right)\to \left|A\right|\in \mathbb{N}$$
6	5	nnge1d	$$⊢\left(\left(A\in V\wedge A\ne \varnothing \right)\wedge A\in \mathrm{Fin}\right)\to 1\le \left|A\right|$$
7		1xr	$$⊢1\in {ℝ}^{*}$$
8		pnfge	$$⊢1\in {ℝ}^{*}\to 1\le \mathrm{+\infty }$$
9	7 8	ax-mp	$$⊢1\le \mathrm{+\infty }$$
10		hashinf	$$⊢\left(A\in V\wedge \neg A\in \mathrm{Fin}\right)\to \left|A\right|=\mathrm{+\infty }$$
11	10	adantlr	$$⊢\left(\left(A\in V\wedge A\ne \varnothing \right)\wedge \neg A\in \mathrm{Fin}\right)\to \left|A\right|=\mathrm{+\infty }$$
12	9 11	breqtrrid	$$⊢\left(\left(A\in V\wedge A\ne \varnothing \right)\wedge \neg A\in \mathrm{Fin}\right)\to 1\le \left|A\right|$$
13	6 12	pm2.61dan	$$⊢\left(A\in V\wedge A\ne \varnothing \right)\to 1\le \left|A\right|$$