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Theorem nzrnz 20746
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 20745 . 2 (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 10 ))
43simprbi 498 1 (𝑅 ∈ NzRing → 10 )
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  wne 2940  cfv 6497  0gc0g 17326  1rcur 19918  Ringcrg 19969  NzRingcnzr 20743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-iota 6449  df-fv 6505  df-nzr 20744
This theorem is referenced by:  nzrunit  20753  subrgnzr  20754  fidomndrng  20794  uvcf1  21214  lindfind2  21240  nm1  24047  deg1pw  25501  ply1nz  25502  ply1nzb  25503  lgsqrlem4  26713  ply1moneq  32335  zrhnm  32607  uvcn0  40773  mon1pid  41575  deg1mhm  41577  nrhmzr  46257
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