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Theorem nzrnz 16294
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 16293 . 2  |-  ( R  e. NzRing 
<->  ( R  e.  Ring  /\  .1.  =/=  .0.  )
)
43simprbi 451 1  |-  ( R  e. NzRing  ->  .1.  =/=  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721    =/= wne 2575   ` cfv 5421   0gc0g 13686   Ringcrg 15623   1rcur 15625  NzRingcnzr 16291
This theorem is referenced by:  nzrunit  16300  subrgnzr  16301  fidomndrng  16330  nm1  18664  deg1pw  20004  ply1nz  20005  ply1nzb  20006  lgsqrlem4  21089  zrhnm  24314  uvcf1  27117  lindfind2  27164  mon1pid  27400  deg1mhm  27402
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-rex 2680  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-iota 5385  df-fv 5429  df-nzr 16292
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