Theorem nzrnz 13364

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 13363	. 2 |- ( R e. NzRing <-> ( R e. Ring /\ .1. =/= .0. ) )
4	3	simprbi 275	1 |- ( R e. NzRing -> .1. =/= .0. )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2148   =/= wne 2347   ` cfv 5218   0gc0g 12711   1rcur 13180   Ringcrg 13217  NzRingcnzr 13361

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-rex 2461  df-rab 2464  df-v 2741  df-un 3135  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-iota 5180  df-fv 5226  df-nzr 13362

This theorem is referenced by:  nzrunit  13367  lringnz  13374  subrgnzr  13401