Theorem nzrnz 20294

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 ∈ NzRing → 1 ≠ 0 )

Proof of Theorem nzrnz

Step	Hyp	Ref	Expression
1		isnzr.o	. . 3 ⊢ 1 = (1r‘𝑅)
2		isnzr.z	. . 3 ⊢ 0 = (0g‘𝑅)
3	1, 2	isnzr 20293	. 2 ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 1 ≠ 0 ))
4	3	simprbi 498	1 ⊢ (𝑅 ∈ NzRing → 1 ≠ 0 )

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2107   ≠ wne 2941  ‘cfv 6544  0_gc0g 17385  1_rcur 20004  Ringcrg 20056  NzRingcnzr 20291

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 2942  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-nzr 20292

This theorem is referenced by:  nzrunit  20301  lringnz  20313  subrgnzr  20341  fidomndrng  20926  uvcf1  21347  lindfind2  21373  nm1  24184  deg1pw  25638  ply1nz  25639  ply1nzb  25640  lgsqrlem4  26852  drngidl  32551  drngidlhash  32552  drnglidl1ne0  32591  drng0mxidl  32592  qsdrngi  32609  ply1moneq  32665  ply1annnr  32764  algextdeglem1  32772  zrhnm  32949  uvcn0  41112  mon1pid  41947  deg1mhm  41949  nrhmzr  46647