Theorem upwordisword 46474

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

Assertion

Ref	Expression
upwordisword	⊢ (𝐴 ∈ UpWord𝑆 → 𝐴 ∈ Word 𝑆)

Proof of Theorem upwordisword

Dummy variables 𝑘 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2828	. 2 ⊢ (𝑤 = 𝐴 → (𝑤 ∈ Word 𝑆 ↔ 𝐴 ∈ Word 𝑆))
2		df-upword 46472	. . . 4 ⊢ UpWord𝑆 = {𝑤 ∣ (𝑤 ∈ Word 𝑆 ∧ ∀𝑘 ∈ (0..^((♯‘𝑤) − 1))(𝑤‘𝑘) < (𝑤‘(𝑘 + 1)))}
3	2	abeq2i 2877	. . 3 ⊢ (𝑤 ∈ UpWord𝑆 ↔ (𝑤 ∈ Word 𝑆 ∧ ∀𝑘 ∈ (0..^((♯‘𝑤) − 1))(𝑤‘𝑘) < (𝑤‘(𝑘 + 1))))
4	3	simplbi 498	. 2 ⊢ (𝑤 ∈ UpWord𝑆 → 𝑤 ∈ Word 𝑆)
5	1, 4	vtoclga 3512	1 ⊢ (𝐴 ∈ UpWord𝑆 → 𝐴 ∈ Word 𝑆)

Syntax hints:   → wi 4   ∧ wa 396   ∈ wcel 2110  ∀wral 3066   class class class wbr 5079  ‘cfv 6431  (class class class)co 7269  0cc0 10864  1c1 10865   + caddc 10867   < clt 11002   − cmin 11197  ..^cfzo 13373  ♯chash 14034  Word cword 14207  UpWordcupword 46471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-12 2175  ax-ext 2711

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1545  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-upword 46472

This theorem is referenced by: (None)