Theorem disjwrdpfx 14891

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 𝑦 ∈ 𝑊 {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

Step	Hyp	Ref	Expression
1		invdisjrab 5095	1 ⊢ Disj 𝑦 ∈ 𝑊 {𝑥 ∈ Word 𝑉 ∣ (𝑥 prefix 𝑁) = 𝑦}

This proof depends on syntax axioms:   = wceq 1570  {crab 3416  Disj wdisj 5075  (class class class)co 7451  Word cword 14699   prefix cpfx 14862

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rmo 3369  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-disj 5076

This theorem is used by:  disjxwwlksn  30550  disjxwwlkn  30559