Theorem nzrring 20488

Description: A nonzero ring is a ring. (Contributed by Stefan O'Rear, 24-Feb-2015.) (Proof shortened by SN, 23-Feb-2025.)

Assertion

Ref	Expression
nzrring	⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring)

Proof of Theorem nzrring

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-nzr 20485	. . 3 ⊢ NzRing = {𝑟 ∈ Ring ∣ (1r‘𝑟) ≠ (0g‘𝑟)}
2	1	ssrab3 4023	. 2 ⊢ NzRing ⊆ Ring
3	2	sseli 3918	1 ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring)

Syntax hints:   → wi 4   ∈ wcel 2114   ≠ wne 2933  ‘cfv 6494  0_gc0g 17397  1_rcur 20157  Ringcrg 20209  NzRingcnzr 20484

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3391  df-ss 3907  df-nzr 20485

This theorem is referenced by:  nzrunit  20496  lringring  20514  rrgnz  20676  domnring  20679  isdomn4  20688  domnchr  21526  uvcf1  21786  lindfind2  21812  frlmisfrlm  21842  nminvr  24648  deg1pw  26100  ply1nz  26101  mon1pid  26133  ply1remlem  26144  ply1rem  26145  facth1  26146  fta1glem1  26147  fta1glem2  26148  unitnz  33319  drngidl  33512  drngidlhash  33513  prmidl0  33529  drnglidl1ne0  33554  drngmxidlr  33557  krull  33558  qsdrngilem  33573  qsdrngi  33574  qsdrnglem2  33575  qsdrng  33576  ply1moneq  33667  deg1vr  33671  zrhnm  34131  abvexp  42995  uvcn0  43005  0prjspnlem  43074  mon1psubm  43649  nzrneg1ne0  48722  islindeps2  48975