ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nzrnz GIF version

Theorem nzrnz 13257
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 13256 . 2 (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 10 ))
43simprbi 275 1 (𝑅 ∈ NzRing → 10 )
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  wcel 2148  wne 2347  cfv 5215  0gc0g 12693  1rcur 13073  Ringcrg 13110  NzRingcnzr 13254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-rex 2461  df-rab 2464  df-v 2739  df-un 3133  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4003  df-iota 5177  df-fv 5223  df-nzr 13255
This theorem is referenced by:  nzrunit  13260  lringnz  13267  subrgnzr  13301
  Copyright terms: Public domain W3C validator