Theorem upwordisword 46735

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 2826	. 2 ⊢ (𝑤 = 𝐴 → (𝑤 ∈ Word 𝑆 ↔ 𝐴 ∈ Word 𝑆))
2		df-upword 46733	. . . 4 ⊢ UpWord 𝑆 = {𝑤 ∣ (𝑤 ∈ Word 𝑆 ∧ ∀𝑘 ∈ (0..^((♯‘𝑤) − 1))(𝑤‘𝑘) < (𝑤‘(𝑘 + 1)))}
3	2	eqabri 2882	. . 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 3585	1 ⊢ (𝐴 ∈ UpWord 𝑆 → 𝐴 ∈ Word 𝑆)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2103  ∀wral 3063   class class class wbr 5169  ‘cfv 6572  (class class class)co 7445  0cc0 11180  1c1 11181   + caddc 11183   < clt 11320   − cmin 11516  ..^cfzo 13707  ♯chash 14375  Word cword 14558  UpWord cupword 46732

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-12 2173  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2712  df-cleq 2726  df-clel 2813  df-upword 46733

This theorem is referenced by:  singoutnupword  46737  upwordsseti  46739  upwrdfi  46741