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Theorem s1nz 14502
Description: A singleton word is not the empty string. (Contributed by Mario Carneiro, 27-Feb-2016.) (Proof shortened by Kyle Wyonch, 18-Jul-2021.)
Assertion
Ref Expression
s1nz ⟨“𝐴”⟩ ≠ ∅

Proof of Theorem s1nz
StepHypRef Expression
1 df-s1 14491 . 2 ⟨“𝐴”⟩ = {⟨0, ( I ‘𝐴)⟩}
2 opex 5426 . . 3 ⟨0, ( I ‘𝐴)⟩ ∈ V
32snnz 4742 . 2 {⟨0, ( I ‘𝐴)⟩} ≠ ∅
41, 3eqnetri 3015 1 ⟨“𝐴”⟩ ≠ ∅
Colors of variables: wff setvar class
Syntax hints:  wne 2944  c0 4287  {csn 4591  cop 4597   I cid 5535  cfv 6501  0cc0 11058  ⟨“cs1 14490
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-s1 14491
This theorem is referenced by:  lswccats1  14529  efgs1  19524  singoutnword  45195
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