Theorem nzrnz 14489

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 14488	. 2 |- ( R e. NzRing <-> ( R e. Ring /\ .1. =/= .0. ) )
4	3	simprbi 275	1 |- ( R e. NzRing -> .1. =/= .0. )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   e. wcel 2209   =/= wne 2420   ` cfv 5377   0gc0g 13610   1rcur 14262   Ringcrg 14300  NzRingcnzr 14486

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-rex 2534  df-rab 2537  df-v 2823  df-un 3224  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-iota 5337  df-fv 5385  df-nzr 14487

This theorem is used by:  nzrunit  14495  lringnz  14502  subrgnzr  14550  rrgnz  14577  drngunz  14618