Theorem nzrnz 20430

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 20429	. 2 ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 1 ≠ 0 ))
4	3	simprbi 496	1 ⊢ (𝑅 ∈ NzRing → 1 ≠ 0 )

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  ‘cfv 6481  0_gc0g 17343  1_rcur 20099  Ringcrg 20151  NzRingcnzr 20427

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-iota 6437  df-fv 6489  df-nzr 20428

This theorem is referenced by:  nzrunit  20439  nrhmzr  20452  lringnz  20458  subrgnzr  20509  rrgnz  20619  fidomndrng  20688  uvcf1  21729  lindfind2  21755  nm1  24582  deg1pw  26053  ply1nz  26054  ply1nzb  26055  mon1pid  26086  lgsqrlem4  27287  unitnz  33206  domnprodn0  33242  fracfld  33274  drngidl  33398  drngidlhash  33399  drnglidl1ne0  33440  drng0mxidl  33441  qsdrngi  33460  ply1moneq  33550  deg1vr  33553  vr1nz  33554  dimlssid  33645  ply1annnr  33716  algextdeglem4  33733  rtelextdg2lem  33739  zrhnm  33980  idomnnzpownz  42235  idomnnzgmulnz  42236  deg1gprod  42243  deg1pow  42244  domnexpgn0cl  42626  abvexp  42635  fiabv  42639  uvcn0  42645  deg1mhm  43303