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Theorem nzrnz 17690
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
StepHypRef Expression
1 isnzr.o . . 3  |-  .1.  =  ( 1r `  R )
2 isnzr.z . . 3  |-  .0.  =  ( 0g `  R )
31, 2isnzr 17689 . 2  |-  ( R  e. NzRing 
<->  ( R  e.  Ring  /\  .1.  =/=  .0.  )
)
43simprbi 464 1  |-  ( R  e. NzRing  ->  .1.  =/=  .0.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767    =/= wne 2662   ` cfv 5586   0gc0g 14691   1rcur 16943   Ringcrg 16986  NzRingcnzr 17687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-iota 5549  df-fv 5594  df-nzr 17688
This theorem is referenced by:  nzrunit  17696  subrgnzr  17697  fidomndrng  17727  uvcf1  18590  lindfind2  18620  nm1  20911  deg1pw  22256  ply1nz  22257  ply1nzb  22258  lgsqrlem4  23347  zrhnm  27586  mon1pid  30770  deg1mhm  30772
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