Theorem nzrnz 13492

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 13491	. 2 |- ( R e. NzRing <-> ( R e. Ring /\ .1. =/= .0. ) )
4	3	simprbi 275	1 |- ( R e. NzRing -> .1. =/= .0. )

Syntax hints:   -> wi 4   = wceq 1363   e. wcel 2159   =/= wne 2359   ` cfv 5230   0gc0g 12726   1rcur 13273   Ringcrg 13310  NzRingcnzr 13489

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2170

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ne 2360  df-rex 2473  df-rab 2476  df-v 2753  df-un 3147  df-sn 3612  df-pr 3613  df-op 3615  df-uni 3824  df-br 4018  df-iota 5192  df-fv 5238  df-nzr 13490

This theorem is referenced by:  nzrunit  13495  lringnz  13502  subrgnzr  13549