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Theorem nzrring 13258
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 13255 . . 3 NzRing = {𝑟 ∈ Ring ∣ (1r𝑟) ≠ (0g𝑟)}
21ssrab3 3241 . 2 NzRing ⊆ Ring
32sseli 3151 1 (𝑅 ∈ NzRing → 𝑅 ∈ Ring)
Colors of variables: wff set class
Syntax hints:  wi 4  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-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-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rab 2464  df-in 3135  df-ss 3142  df-nzr 13255
This theorem is referenced by:  nzrunit  13260  lringring  13266
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