Theorem s1nz 13386

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 13302	. 2 |- <" A "> = { <. 0 , ( _I ` A ) >. }
2		opex 4932	. . 3 |- <. 0 , ( _I ` A ) >. e. _V
3	2	snnz 4309	. 2 |- { <. 0 , ( _I ` A ) >. } =/= (/)
4	1, 3	eqnetri 2864	1 |- <" A "> =/= (/)

Syntax hints:   =/= wne 2794   (/)c0 3915   {csn 4177   <.cop 4183   _I cid 5023   ` cfv 5888   0cc0 9936   <"cs1 13294

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-s1 13302

This theorem is referenced by:  lswccats1  13411  efgs1  18148