Theorem upwordisword 45595

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 2822	. 2 ⊢ (𝑤 = 𝐴 → (𝑤 ∈ Word 𝑆 ↔ 𝐴 ∈ Word 𝑆))
2		df-upword 45593	. . . 4 ⊢ UpWord 𝑆 = {𝑤 ∣ (𝑤 ∈ Word 𝑆 ∧ ∀𝑘 ∈ (0..^((♯‘𝑤) − 1))(𝑤‘𝑘) < (𝑤‘(𝑘 + 1)))}
3	2	eqabri 2878	. . 3 ⊢ (𝑤 ∈ UpWord 𝑆 ↔ (𝑤 ∈ Word 𝑆 ∧ ∀𝑘 ∈ (0..^((♯‘𝑤) − 1))(𝑤‘𝑘) < (𝑤‘(𝑘 + 1))))
4	3	simplbi 499	. 2 ⊢ (𝑤 ∈ UpWord 𝑆 → 𝑤 ∈ Word 𝑆)
5	1, 4	vtoclga 3566	1 ⊢ (𝐴 ∈ UpWord 𝑆 → 𝐴 ∈ Word 𝑆)

Syntax hints:   → wi 4   ∧ wa 397   ∈ wcel 2107  ∀wral 3062   class class class wbr 5149  ‘cfv 6544  (class class class)co 7409  0cc0 11110  1c1 11111   + caddc 11113   < clt 11248   − cmin 11444  ..^cfzo 13627  ♯chash 14290  Word cword 14464  UpWord cupword 45592

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-12 2172  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-upword 45593

This theorem is referenced by:  singoutnupword  45597  upwordsseti  45599  upwrdfi  45601