MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  s1nz Structured version   Visualization version   GIF version

Theorem s1nz 14583
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 14572 . 2 ⟨“𝐴”⟩ = {⟨0, ( I ‘𝐴)⟩}
2 opex 5460 . . 3 ⟨0, ( I ‘𝐴)⟩ ∈ V
32snnz 4776 . 2 {⟨0, ( I ‘𝐴)⟩} ≠ ∅
41, 3eqnetri 3007 1 ⟨“𝐴”⟩ ≠ ∅
Colors of variables: wff setvar class
Syntax hints:  wne 2936  c0 4318  {csn 4624  cop 4630   I cid 5569  cfv 6542  0cc0 11132  ⟨“cs1 14571
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2937  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-s1 14572
This theorem is referenced by:  lswccats1  14610  efgs1  19683  singoutnword  46262
  Copyright terms: Public domain W3C validator