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Theorem nzrring 13389
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.
StepHypRef Expression
1 df-nzr 13386 . . 3 NzRing = {𝑟 ∈ Ring ∣ (1r𝑟) ≠ (0g𝑟)}
21ssrab3 3253 . 2 NzRing ⊆ Ring
32sseli 3163 1 (𝑅 ∈ NzRing → 𝑅 ∈ Ring)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 2158  wne 2357  cfv 5228  0gc0g 12722  1rcur 13196  Ringcrg 13233  NzRingcnzr 13385
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-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-rab 2474  df-in 3147  df-ss 3154  df-nzr 13386
This theorem is referenced by:  nzrunit  13391  lringring  13397
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