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

Theorem disjwrdpfx 14151
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 5016 1 Disj 𝑦𝑊 {𝑥 ∈ Word 𝑉 ∣ (𝑥 prefix 𝑁) = 𝑦}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  {crab 3057  Disj wdisj 4995  (class class class)co 7170  Word cword 13955   prefix cpfx 14121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ral 3058  df-rmo 3061  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-disj 4996
This theorem is referenced by:  disjxwwlksn  27842  disjxwwlkn  27851
  Copyright terms: Public domain W3C validator