Theorem upwordisword 46872

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

Assertion

Ref	Expression
upwordisword	⊢ (𝐴 ∈ UpWord 𝑆 → 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 2817	. 2 ⊢ (𝑤 = 𝐴 → (𝑤 ∈ Word 𝑆 ↔ 𝐴 ∈ Word 𝑆))
2		df-upword 46870	. . . 4 ⊢ UpWord 𝑆 = {𝑤 ∣ (𝑤 ∈ Word 𝑆 ∧ ∀𝑘 ∈ (0..^((♯‘𝑤) − 1))(𝑤‘𝑘) < (𝑤‘(𝑘 + 1)))}
3	2	eqabri 2872	. . 3 ⊢ (𝑤 ∈ UpWord 𝑆 ↔ (𝑤 ∈ Word 𝑆 ∧ ∀𝑘 ∈ (0..^((♯‘𝑤) − 1))(𝑤‘𝑘) < (𝑤‘(𝑘 + 1))))
4	3	simplbi 497	. 2 ⊢ (𝑤 ∈ UpWord 𝑆 → 𝑤 ∈ Word 𝑆)
5	1, 4	vtoclga 3546	1 ⊢ (𝐴 ∈ UpWord 𝑆 → 𝐴 ∈ Word 𝑆)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2109  ∀wral 3045   class class class wbr 5109  ‘cfv 6513  (class class class)co 7389  0cc0 11074  1c1 11075   + caddc 11077   < clt 11214   − cmin 11411  ..^cfzo 13621  ♯chash 14301  Word cword 14484  UpWord cupword 46869

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-12 2178  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-upword 46870

This theorem is referenced by:  singoutnupword  46874  upwordsseti  46876  upwrdfi  46878