Theorem nzrnz 19260

Description: One and zero are different in a nonzero ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)

Hypotheses

Ref	Expression
isnzr.o	|- .1. = ( 1r ` R )
isnzr.z	|- .0. = ( 0g ` R )

Assertion

Ref	Expression
nzrnz	|- ( R e. NzRing -> .1. =/= .0. )

Proof of Theorem nzrnz

Step	Hyp	Ref	Expression
1		isnzr.o	. . 3 |- .1. = ( 1r ` R )
2		isnzr.z	. . 3 |- .0. = ( 0g ` R )
3	1, 2	isnzr 19259	. 2 |- ( R e. NzRing <-> ( R e. Ring /\ .1. =/= .0. ) )
4	3	simprbi 480	1 |- ( R e. NzRing -> .1. =/= .0. )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   =/= wne 2794   ` cfv 5888   0gc0g 16100   1rcur 18501   Ringcrg 18547  NzRingcnzr 19257

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

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-rex 2918  df-rab 2921  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-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-nzr 19258

This theorem is referenced by:  nzrunit  19267  subrgnzr  19268  fidomndrng  19307  uvcf1  20131  lindfind2  20157  nm1  22471  deg1pw  23880  ply1nz  23881  ply1nzb  23882  lgsqrlem4  25074  zrhnm  30013  mon1pid  37783  deg1mhm  37785  nrhmzr  41873