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Theorem disjxwrd 13307
Description: Sets of words are disjoint if each set contains extensions of distinct words of a fixed length. (Contributed by AV, 29-Jul-2018.) (Proof shortened by AV, 7-May-2020.)
Assertion
Ref Expression
disjxwrd Disj 𝑦𝑊 {𝑥 ∈ Word 𝑉 ∣ (𝑥 substr ⟨0, 𝑁⟩) = 𝑦}
Distinct variable groups:   𝑦,𝑁   𝑥,𝑉   𝑥,𝑦
Allowed substitution hints:   𝑁(𝑥)   𝑉(𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem disjxwrd
StepHypRef Expression
1 invdisjrab 4572 1 Disj 𝑦𝑊 {𝑥 ∈ Word 𝑉 ∣ (𝑥 substr ⟨0, 𝑁⟩) = 𝑦}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1475  {crab 2900  cop 4131  Disj wdisj 4553  (class class class)co 6549  0cc0 9815  Word cword 13146   substr csubstr 13150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-disj 4554
This theorem is referenced by:  disjxwwlks  26264  disjxwwlkn  26273  disjxwwlksn  41110  av-disjxwwlkn  41119
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