Theorem chnpolfz

Description: Provided that chain's relation is a partial order, the chain length is restricted to a specific integer range. (Contributed by Ender Ting, 20-Jan-2026)

	Ref	Expression
Hypotheses	chnpolfz.1	$$⊢\phi \to \underset{\dot{}}{<}\mathrm{Po}A$$
	chnpolfz.2	$$⊢\phi \to B\in {\mathrm{Chain}}_A\underset{\dot{}}{<}$$
	chnpolfz.3	$$⊢\phi \to A\in \mathrm{Fin}$$
Assertion	chnpolfz	$$⊢\phi \to \left|B\right|\in \left(0\dots \left|A\right|\right)$$

Proof

Step	Hyp	Ref	Expression
1		chnpolfz.1	$$⊢\phi \to \underset{\dot{}}{<}\mathrm{Po}A$$
2		chnpolfz.2	$$⊢\phi \to B\in {\mathrm{Chain}}_A\underset{\dot{}}{<}$$
3		chnpolfz.3	$$⊢\phi \to A\in \mathrm{Fin}$$
4		0zd	$$⊢\phi \to 0\in \mathbb{Z}$$
5		hashcl	$$⊢A\in \mathrm{Fin}\to \left|A\right|\in {\mathbb{N}}_0$$
6	3 5	syl	$$⊢\phi \to \left|A\right|\in {\mathbb{N}}_0$$
7	6	nn0zd	$$⊢\phi \to \left|A\right|\in \mathbb{Z}$$
8	2	chnwrd	$$⊢\phi \to B\in \mathrm{Word}A$$
9		lencl	$$⊢B\in \mathrm{Word}A\to \left|B\right|\in {\mathbb{N}}_0$$
10	8 9	syl	$$⊢\phi \to \left|B\right|\in {\mathbb{N}}_0$$
11	10	nn0zd	$$⊢\phi \to \left|B\right|\in \mathbb{Z}$$
12		hashge0	$$⊢B\in {\mathrm{Chain}}_A\underset{\dot{}}{<}\to 0\le \left|B\right|$$
13	2 12	syl	$$⊢\phi \to 0\le \left|B\right|$$
14	1 2 3	chnpolleha	$$⊢\phi \to \left|B\right|\le \left|A\right|$$
15	4 7 11 13 14	elfzd	$$⊢\phi \to \left|B\right|\in \left(0\dots \left|A\right|\right)$$