Theorem chnwrd

Description: A chain is an ordered sequence, i.e. a word. (Contributed by Thierry Arnoux, 19-Jun-2025)

	Ref	Expression
Hypothesis	chnwrd.1	⊢ ( 𝜑 → 𝐶 ∈ ( < Chain 𝐴 ) )
Assertion	chnwrd	⊢ ( 𝜑 → 𝐶 ∈ Word 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		chnwrd.1	⊢ ( 𝜑 → 𝐶 ∈ ( < Chain 𝐴 ) )
2		ischn	⊢ ( 𝐶 ∈ ( < Chain 𝐴 ) ↔ ( 𝐶 ∈ Word 𝐴 ∧ ∀ 𝑛 ∈ ( dom 𝐶 ∖ { 0 } ) ( 𝐶 ‘ ( 𝑛 − 1 ) ) < ( 𝐶 ‘ 𝑛 ) ) )
3	2	simplbi	⊢ ( 𝐶 ∈ ( < Chain 𝐴 ) → 𝐶 ∈ Word 𝐴 )
4	1 3	syl	⊢ ( 𝜑 → 𝐶 ∈ Word 𝐴 )