Theorem s1nz 13697

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 13686	. 2 ⊢ ⟨“𝐴”⟩ = {⟨0, ( I ‘𝐴)⟩}
2		opex 5164	. . 3 ⊢ ⟨0, ( I ‘𝐴)⟩ ∈ V
3	2	snnz 4541	. 2 ⊢ {⟨0, ( I ‘𝐴)⟩} ≠ ∅
4	1, 3	eqnetri 3038	1 ⊢ ⟨“𝐴”⟩ ≠ ∅

Syntax hints:   ≠ wne 2968  ∅c0 4140  {csn 4397  ⟨cop 4403   I cid 5260  ‘cfv 6135  0cc0 10272  ⟨“cs1 13685

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pr 5138

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-v 3399  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-s1 13686

This theorem is referenced by:  lswccats1  13724  efgs1  18532