Theorem s1nz 14578

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 14567	. 2 ⊢ ⟨“𝐴”⟩ = {⟨0, ( I ‘𝐴)⟩}
2		opex 5426	. . 3 ⊢ ⟨0, ( I ‘𝐴)⟩ ∈ V
3	2	snnz 4742	. 2 ⊢ {⟨0, ( I ‘𝐴)⟩} ≠ ∅
4	1, 3	eqnetri 2996	1 ⊢ ⟨“𝐴”⟩ ≠ ∅

Syntax hints:   ≠ wne 2926  ∅c0 4298  {csn 4591  ⟨cop 4597   I cid 5534  ‘cfv 6513  0cc0 11074  ⟨“cs1 14566

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5253  ax-nul 5263  ax-pr 5389

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-v 3452  df-dif 3919  df-un 3921  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-op 4598  df-s1 14567

This theorem is referenced by:  lswccats1  14605  efgs1  19671  singoutnword  46873