Theorem lennncl

Description: The length of a nonempty word is a positive integer. (Contributed by Mario Carneiro, 1-Oct-2015)

	Ref	Expression
Assertion	lennncl	$$⊢\left(W\in \mathrm{Word}S\wedge W\ne \varnothing \right)\to \left|W\right|\in \mathbb{N}$$

Proof

Step	Hyp	Ref	Expression
1		wrdfin	$$⊢W\in \mathrm{Word}S\to W\in \mathrm{Fin}$$
2		hashnncl	$$⊢W\in \mathrm{Fin}\to \left(\left|W\right|\in \mathbb{N}\leftrightarrow W\ne \varnothing \right)$$
3	1 2	syl	$$⊢W\in \mathrm{Word}S\to \left(\left|W\right|\in \mathbb{N}\leftrightarrow W\ne \varnothing \right)$$
4	3	biimpar	$$⊢\left(W\in \mathrm{Word}S\wedge W\ne \varnothing \right)\to \left|W\right|\in \mathbb{N}$$