Theorem s1nz 14758

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 14747	. 2 ⊢ ⟨“𝐴”⟩ = {⟨0, ( I ‘𝐴)⟩}
2		opex 5431	. . 3 ⊢ ⟨0, ( I ‘𝐴)⟩ ∈ V
3	2	snnz 4733	. 2 ⊢ {⟨0, ( I ‘𝐴)⟩} ≠ ∅
4	1, 3	eqnetri 3025	1 ⊢ ⟨“𝐴”⟩ ≠ ∅

This proof depends on syntax axioms:   ≠ wne 2955  ∅c0 4278  {csn 4584  ⟨cop 4834   I cid 5544  ‘cfv 6539  0cc0 11204  ⟨“cs1 14746

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-sn 4585  df-pr 4587  df-op 4835  df-s1 14747

This theorem is used by:  lswccats1  14787  efgs1  19955