Theorem s1nz 14512

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

Step	Hyp	Ref	Expression
1		df-s1 14501	. 2 ⊢ ⟨“𝐴”⟩ = {⟨0, ( I ‘𝐴)⟩}
2		opex 5404	. . 3 ⊢ ⟨0, ( I ‘𝐴)⟩ ∈ V
3	2	snnz 4729	. 2 ⊢ {⟨0, ( I ‘𝐴)⟩} ≠ ∅
4	1, 3	eqnetri 2998	1 ⊢ ⟨“𝐴”⟩ ≠ ∅

Syntax hints:   ≠ wne 2928  ∅c0 4283  {csn 4576  ⟨cop 4582   I cid 5510  ‘cfv 6481  0cc0 11003  ⟨“cs1 14500

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-v 3438  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-s1 14501

This theorem is referenced by:  lswccats1  14539  efgs1  19645