Theorem s1nz 14625

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 14614	. 2 ⊢ ⟨“𝐴”⟩ = {⟨0, ( I ‘𝐴)⟩}
2		opex 5439	. . 3 ⊢ ⟨0, ( I ‘𝐴)⟩ ∈ V
3	2	snnz 4752	. 2 ⊢ {⟨0, ( I ‘𝐴)⟩} ≠ ∅
4	1, 3	eqnetri 3002	1 ⊢ ⟨“𝐴”⟩ ≠ ∅

Syntax hints:   ≠ wne 2932  ∅c0 4308  {csn 4601  ⟨cop 4607   I cid 5547  ‘cfv 6531  0cc0 11129  ⟨“cs1 14613

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 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

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 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ne 2933  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-s1 14614

This theorem is referenced by:  lswccats1  14652  efgs1  19716  singoutnword  46911