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Theorem nzrnz 19475
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 19474 . 2 (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 10 ))
43simprbi 484 1 (𝑅 ∈ NzRing → 10 )
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  wne 2943  cfv 6031  0gc0g 16308  1rcur 18709  Ringcrg 18755  NzRingcnzr 19472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-iota 5994  df-fv 6039  df-nzr 19473
This theorem is referenced by:  nzrunit  19482  subrgnzr  19483  fidomndrng  19522  uvcf1  20348  lindfind2  20374  nm1  22691  deg1pw  24100  ply1nz  24101  ply1nzb  24102  lgsqrlem4  25295  zrhnm  30353  mon1pid  38309  deg1mhm  38311  nrhmzr  42401
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