Theorem upwordisword

Description: Any increasing sequence is a sequence. (Contributed by Ender Ting, 19-Nov-2024)

	Ref	Expression
Assertion	upwordisword	⊢ ( 𝐴 ∈ UpWord 𝑆 → 𝐴 ∈ Word 𝑆 )

Proof

Step	Hyp	Ref	Expression
1		eleq1	⊢ ( 𝑤 = 𝐴 → ( 𝑤 ∈ Word 𝑆 ↔ 𝐴 ∈ Word 𝑆 ) )
2		df-upword	⊢ UpWord 𝑆 = { 𝑤 ∣ ( 𝑤 ∈ Word 𝑆 ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑤 ) − 1 ) ) ( 𝑤 ‘ 𝑘 ) < ( 𝑤 ‘ ( 𝑘 + 1 ) ) ) }
3	2	abeq2i	⊢ ( 𝑤 ∈ UpWord 𝑆 ↔ ( 𝑤 ∈ Word 𝑆 ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑤 ) − 1 ) ) ( 𝑤 ‘ 𝑘 ) < ( 𝑤 ‘ ( 𝑘 + 1 ) ) ) )
4	3	simplbi	⊢ ( 𝑤 ∈ UpWord 𝑆 → 𝑤 ∈ Word 𝑆 )
5	1 4	vtoclga	⊢ ( 𝐴 ∈ UpWord 𝑆 → 𝐴 ∈ Word 𝑆 )