Theorem nulchn

Description: Empty set is an increasing chain for every range and every relation. (Contributed by Ender Ting, 19-Nov-2024) (Revised by Ender Ting, 17-Jan-2026)

	Ref	Expression
Assertion	nulchn	$$⊢\varnothing \in {\mathrm{Chain}}_A\underset{\dot{}}{<}$$

Proof

Step	Hyp	Ref	Expression
1		wrd0	$$⊢\varnothing \in \mathrm{Word}A$$
2		dm0	$$⊢\mathrm{dom}\varnothing =\varnothing$$
3	2	difeq1i	$$⊢\mathrm{dom}\varnothing \setminus \left\{0\right\}=\varnothing \setminus \left\{0\right\}$$
4		0dif	$$⊢\varnothing \setminus \left\{0\right\}=\varnothing$$
5	3 4	eqtri	$$⊢\mathrm{dom}\varnothing \setminus \left\{0\right\}=\varnothing$$
6		rzal	$$⊢\mathrm{dom}\varnothing \setminus \left\{0\right\}=\varnothing \to \forall x\in \left(\mathrm{dom}\varnothing \setminus \left\{0\right\}\right)\varnothing \left(x-1\right)\underset{\dot{}}{<}\varnothing \left(x\right)$$
7	5 6	ax-mp	$$⊢\forall x\in \left(\mathrm{dom}\varnothing \setminus \left\{0\right\}\right)\varnothing \left(x-1\right)\underset{\dot{}}{<}\varnothing \left(x\right)$$
8	1 7	pm3.2i	$$⊢\left(\varnothing \in \mathrm{Word}A\wedge \forall x\in \left(\mathrm{dom}\varnothing \setminus \left\{0\right\}\right)\varnothing \left(x-1\right)\underset{\dot{}}{<}\varnothing \left(x\right)\right)$$
9		ischn	$$⊢\varnothing \in {\mathrm{Chain}}_A\underset{\dot{}}{<}\leftrightarrow \left(\varnothing \in \mathrm{Word}A\wedge \forall x\in \left(\mathrm{dom}\varnothing \setminus \left\{0\right\}\right)\varnothing \left(x-1\right)\underset{\dot{}}{<}\varnothing \left(x\right)\right)$$
10	8 9	mpbir	$$⊢\varnothing \in {\mathrm{Chain}}_A\underset{\dot{}}{<}$$