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Theorem s1nz 13239
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 13157 . 2 ⟨“𝐴”⟩ = {⟨0, ( I ‘𝐴)⟩}
2 opex 4859 . . 3 ⟨0, ( I ‘𝐴)⟩ ∈ V
32snnz 4252 . 2 {⟨0, ( I ‘𝐴)⟩} ≠ ∅
41, 3eqnetri 2852 1 ⟨“𝐴”⟩ ≠ ∅
Colors of variables: wff setvar class
Syntax hints:  wne 2780  c0 3874  {csn 4125  cop 4131   I cid 4948  cfv 5804  0cc0 9815  ⟨“cs1 13149
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-s1 13157
This theorem is referenced by:  lswccats1  13263  efgs1  17971
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