Theorem chndun

Description: Chains in either of two alphabets are chains in their union. (Contributed by Ender Ting, 24-Jul-2026)

	Ref	Expression
Assertion	chndun	⊢ ( ( 𝐴 ∈ ( 𝑅 Chain 𝐵 ) ∨ 𝐴 ∈ ( 𝑅 Chain 𝐶 ) ) → 𝐴 ∈ ( 𝑅 Chain ( 𝐵 ∪ 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ssun1	⊢ 𝐵 ⊆ ( 𝐵 ∪ 𝐶 )
2		chndss	⊢ ( 𝐵 ⊆ ( 𝐵 ∪ 𝐶 ) → ( 𝑅 Chain 𝐵 ) ⊆ ( 𝑅 Chain ( 𝐵 ∪ 𝐶 ) ) )
3	1 2	ax-mp	⊢ ( 𝑅 Chain 𝐵 ) ⊆ ( 𝑅 Chain ( 𝐵 ∪ 𝐶 ) )
4	3	sseli	⊢ ( 𝐴 ∈ ( 𝑅 Chain 𝐵 ) → 𝐴 ∈ ( 𝑅 Chain ( 𝐵 ∪ 𝐶 ) ) )
5		ssun2	⊢ 𝐶 ⊆ ( 𝐵 ∪ 𝐶 )
6		chndss	⊢ ( 𝐶 ⊆ ( 𝐵 ∪ 𝐶 ) → ( 𝑅 Chain 𝐶 ) ⊆ ( 𝑅 Chain ( 𝐵 ∪ 𝐶 ) ) )
7	5 6	ax-mp	⊢ ( 𝑅 Chain 𝐶 ) ⊆ ( 𝑅 Chain ( 𝐵 ∪ 𝐶 ) )
8	7	sseli	⊢ ( 𝐴 ∈ ( 𝑅 Chain 𝐶 ) → 𝐴 ∈ ( 𝑅 Chain ( 𝐵 ∪ 𝐶 ) ) )
9	4 8	jaoi	⊢ ( ( 𝐴 ∈ ( 𝑅 Chain 𝐵 ) ∨ 𝐴 ∈ ( 𝑅 Chain 𝐶 ) ) → 𝐴 ∈ ( 𝑅 Chain ( 𝐵 ∪ 𝐶 ) ) )