Theorem chnrdss

Description: Subset theorem for chains. (Contributed by Ender Ting, 20-Jan-2026)

	Ref	Expression
Assertion	chnrdss	⊢ ( ( < ⊆ 𝑅 ∧ 𝐴 ⊆ 𝐵 ) → ( < Chain 𝐴 ) ⊆ ( 𝑅 Chain 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		chnrss	⊢ ( < ⊆ 𝑅 → ( < Chain 𝐴 ) ⊆ ( 𝑅 Chain 𝐴 ) )
2		chndss	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝑅 Chain 𝐴 ) ⊆ ( 𝑅 Chain 𝐵 ) )
3		sstr	⊢ ( ( ( < Chain 𝐴 ) ⊆ ( 𝑅 Chain 𝐴 ) ∧ ( 𝑅 Chain 𝐴 ) ⊆ ( 𝑅 Chain 𝐵 ) ) → ( < Chain 𝐴 ) ⊆ ( 𝑅 Chain 𝐵 ) )
4	1 2 3	syl2an	⊢ ( ( < ⊆ 𝑅 ∧ 𝐴 ⊆ 𝐵 ) → ( < Chain 𝐴 ) ⊆ ( 𝑅 Chain 𝐵 ) )