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Theorem disjwrdpfx 14647
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 5124 1 Disj 𝑦𝑊 {𝑥 ∈ Word 𝑉 ∣ (𝑥 prefix 𝑁) = 𝑦}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  {crab 3424  Disj wdisj 5103  (class class class)co 7401  Word cword 14461   prefix cpfx 14617
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-rmo 3368  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-disj 5104
This theorem is referenced by:  disjxwwlksn  29627  disjxwwlkn  29636
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