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Theorem nzrnz 19200
 Description: One and zero are different in a nonzero ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)
Hypotheses
Ref Expression
isnzr.o 1 = (1r𝑅)
isnzr.z 0 = (0g𝑅)
Assertion
Ref Expression
nzrnz (𝑅 ∈ NzRing → 10 )

Proof of Theorem nzrnz
StepHypRef Expression
1 isnzr.o . . 3 1 = (1r𝑅)
2 isnzr.z . . 3 0 = (0g𝑅)
31, 2isnzr 19199 . 2 (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 10 ))
43simprbi 480 1 (𝑅 ∈ NzRing → 10 )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1480   ∈ wcel 1987   ≠ wne 2790  ‘cfv 5857  0gc0g 16040  1rcur 18441  Ringcrg 18487  NzRingcnzr 19197 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-iota 5820  df-fv 5865  df-nzr 19198 This theorem is referenced by:  nzrunit  19207  subrgnzr  19208  fidomndrng  19247  uvcf1  20071  lindfind2  20097  nm1  22411  deg1pw  23818  ply1nz  23819  ply1nzb  23820  lgsqrlem4  25008  zrhnm  29837  mon1pid  37303  deg1mhm  37305  nrhmzr  41191
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