Theorem nzrnz 14260

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 14259	. 2 |- ( R e. NzRing <-> ( R e. Ring /\ .1. =/= .0. ) )
4	3	simprbi 275	1 |- ( R e. NzRing -> .1. =/= .0. )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2202   =/= wne 2403   ` cfv 5333   0gc0g 13402   1rcur 14036   Ringcrg 14073  NzRingcnzr 14257

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-rex 2517  df-rab 2520  df-v 2805  df-un 3205  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-iota 5293  df-fv 5341  df-nzr 14258

This theorem is referenced by:  nzrunit  14266  lringnz  14273  subrgnzr  14320  rrgnz  14347