Theorem chndun2

Description: Superaddivity of chain constructor over alphabet parameter. (Contributed by Ender Ting, 24-Jul-2026)

	Ref	Expression
Assertion	chndun2	$$⊢{\mathrm{Chain}}_{B}R\cup {\mathrm{Chain}}_{C}R\subseteq {\mathrm{Chain}}_{\left(B\cup C\right)}R$$

Proof

Step	Hyp	Ref	Expression
1		elun	$$⊢n\in \left({\mathrm{Chain}}_{B}R\cup {\mathrm{Chain}}_{C}R\right)\leftrightarrow \left(n\in {\mathrm{Chain}}_{B}R\vee n\in {\mathrm{Chain}}_{C}R\right)$$
2		chndun	$$⊢\left(n\in {\mathrm{Chain}}_{B}R\vee n\in {\mathrm{Chain}}_{C}R\right)\to n\in {\mathrm{Chain}}_{\left(B\cup C\right)}R$$
3	1 2	sylbi	$$⊢n\in \left({\mathrm{Chain}}_{B}R\cup {\mathrm{Chain}}_{C}R\right)\to n\in {\mathrm{Chain}}_{\left(B\cup C\right)}R$$
4	3	ssriv	$$⊢{\mathrm{Chain}}_{B}R\cup {\mathrm{Chain}}_{C}R\subseteq {\mathrm{Chain}}_{\left(B\cup C\right)}R$$