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Theorem nzrnz 18228
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 18227 . 2  |-  ( R  e. NzRing 
<->  ( R  e.  Ring  /\  .1.  =/=  .0.  )
)
43simprbi 462 1  |-  ( R  e. NzRing  ->  .1.  =/=  .0.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1405    e. wcel 1842    =/= wne 2598   ` cfv 5569   0gc0g 15054   1rcur 17473   Ringcrg 17518  NzRingcnzr 18225
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-rex 2760  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-iota 5533  df-fv 5577  df-nzr 18226
This theorem is referenced by:  nzrunit  18235  subrgnzr  18236  fidomndrng  18276  uvcf1  19119  lindfind2  19145  nm1  21468  deg1pw  22813  ply1nz  22814  ply1nzb  22815  lgsqrlem4  24000  zrhnm  28402  mon1pid  35529  deg1mhm  35531  nrhmzr  38190
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