MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  disjwrdpfx Structured version   Visualization version   GIF version

Theorem disjwrdpfx 14413
Description: Sets of words are disjoint if each set contains exactly the extensions of distinct words of a fixed length. Remark: A word 𝑊 is called an "extension" of a word 𝑃 if 𝑃 is a prefix of 𝑊. (Contributed by AV, 29-Jul-2018.) (Revised by AV, 6-May-2020.)
Assertion
Ref Expression
disjwrdpfx Disj 𝑦𝑊 {𝑥 ∈ Word 𝑉 ∣ (𝑥 prefix 𝑁) = 𝑦}
Distinct variable groups:   𝑦,𝑁   𝑥,𝑉   𝑥,𝑦
Allowed substitution hints:   𝑁(𝑥)   𝑉(𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem disjwrdpfx
StepHypRef Expression
1 invdisjrab 5060 1 Disj 𝑦𝑊 {𝑥 ∈ Word 𝑉 ∣ (𝑥 prefix 𝑁) = 𝑦}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  {crab 3068  Disj wdisj 5039  (class class class)co 7275  Word cword 14217   prefix cpfx 14383
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rmo 3071  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-disj 5040
This theorem is referenced by:  disjxwwlksn  28269  disjxwwlkn  28278
  Copyright terms: Public domain W3C validator