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Theorem nzrnz 16222
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 16221 . 2  |-  ( R  e. NzRing 
<->  ( R  e.  Ring  /\  .1.  =/=  .0.  )
)
43simprbi 450 1  |-  ( R  e. NzRing  ->  .1.  =/=  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1647    e. wcel 1715    =/= wne 2529   ` cfv 5358   0gc0g 13610   Ringcrg 15547   1rcur 15549  NzRingcnzr 16219
This theorem is referenced by:  nzrunit  16228  subrgnzr  16229  fidomndrng  16258  nm1  18391  deg1pw  19721  ply1nz  19722  ply1nzb  19723  lgsqrlem4  20806  zrhnm  23948  uvcf1  26832  lindfind2  26879  mon1pid  27115  deg1mhm  27117
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-rex 2634  df-rab 2637  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-iota 5322  df-fv 5366  df-nzr 16220
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