Theorem chnrun2

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

	Ref	Expression
Assertion	chnrun2	$$⊢{\mathrm{Chain}}_{B}R\cup {\mathrm{Chain}}_B\underset{\dot{}}{<}\subseteq {\mathrm{Chain}}_B\left(R\cup \underset{\dot{}}{<}\right)$$

Proof

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