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Theorem isnzr 16330
 Description: Property of a nonzero ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)
Hypotheses
Ref Expression
isnzr.o
isnzr.z
Assertion
Ref Expression
isnzr NzRing

Proof of Theorem isnzr
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 fveq2 5728 . . . 4
2 isnzr.o . . . 4
31, 2syl6eqr 2486 . . 3
4 fveq2 5728 . . . 4
5 isnzr.z . . . 4
64, 5syl6eqr 2486 . . 3
73, 6neeq12d 2616 . 2
8 df-nzr 16329 . 2 NzRing
97, 8elrab2 3094 1 NzRing
 Colors of variables: wff set class Syntax hints:   wb 177   wa 359   wceq 1652   wcel 1725   wne 2599  cfv 5454  c0g 13723  crg 15660  cur 15662  NzRingcnzr 16328 This theorem is referenced by:  nzrnz  16331  nzrrng  16332  drngnzr  16333  isnzr2  16334  rngelnzr  16336  subrgnzr  16338  chrnzr  16811  nrginvrcn  18727  ply1nzb  20045  zrhnm  24353  isdomn3  27500 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-nzr 16329
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