Theorem upwordisword 45893

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 2819	. 2 ⊢ (𝑤 = 𝐴 → (𝑤 ∈ Word 𝑆 ↔ 𝐴 ∈ Word 𝑆))
2		df-upword 45891	. . . 4 ⊢ UpWord 𝑆 = {𝑤 ∣ (𝑤 ∈ Word 𝑆 ∧ ∀𝑘 ∈ (0..^((♯‘𝑤) − 1))(𝑤‘𝑘) < (𝑤‘(𝑘 + 1)))}
3	2	eqabri 2875	. . 3 ⊢ (𝑤 ∈ UpWord 𝑆 ↔ (𝑤 ∈ Word 𝑆 ∧ ∀𝑘 ∈ (0..^((♯‘𝑤) − 1))(𝑤‘𝑘) < (𝑤‘(𝑘 + 1))))
4	3	simplbi 496	. 2 ⊢ (𝑤 ∈ UpWord 𝑆 → 𝑤 ∈ Word 𝑆)
5	1, 4	vtoclga 3565	1 ⊢ (𝐴 ∈ UpWord 𝑆 → 𝐴 ∈ Word 𝑆)

Syntax hints:   → wi 4   ∧ wa 394   ∈ wcel 2104  ∀wral 3059   class class class wbr 5147  ‘cfv 6542  (class class class)co 7411  0cc0 11112  1c1 11113   + caddc 11115   < clt 11252   − cmin 11448  ..^cfzo 13631  ♯chash 14294  Word cword 14468  UpWord cupword 45890

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-12 2169  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-upword 45891

This theorem is referenced by:  singoutnupword  45895  upwordsseti  45897  upwrdfi  45899