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Theorem s1nz 13961
 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 13950 . 2 ⟨“𝐴”⟩ = {⟨0, ( I ‘𝐴)⟩}
2 opex 5343 . . 3 ⟨0, ( I ‘𝐴)⟩ ∈ V
32snnz 4696 . 2 {⟨0, ( I ‘𝐴)⟩} ≠ ∅
41, 3eqnetri 3084 1 ⟨“𝐴”⟩ ≠ ∅
 Colors of variables: wff setvar class Syntax hints:   ≠ wne 3014  ∅c0 4276  {csn 4550  ⟨cop 4556   I cid 5446  ‘cfv 6343  0cc0 10535  ⟨“cs1 13949 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-s1 13950 This theorem is referenced by:  lswccats1  13993  efgs1  18861
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