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Theorem nzrring 13494
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 13491 . . 3 NzRing = {𝑟 ∈ Ring ∣ (1r𝑟) ≠ (0g𝑟)}
21ssrab3 3256 . 2 NzRing ⊆ Ring
32sseli 3166 1 (𝑅 ∈ NzRing → 𝑅 ∈ Ring)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 2160  wne 2360  cfv 5231  0gc0g 12727  1rcur 13274  Ringcrg 13311  NzRingcnzr 13490
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rab 2477  df-in 3150  df-ss 3157  df-nzr 13491
This theorem is referenced by:  nzrunit  13496  lringring  13502
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