Theorem nzrring 14196

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 14193	. . 3 ⊢ NzRing = {𝑟 ∈ Ring ∣ (1r‘𝑟) ≠ (0g‘𝑟)}
2	1	ssrab3 3313	. 2 ⊢ NzRing ⊆ Ring
3	2	sseli 3223	1 ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring)

Syntax hints:   → wi 4   ∈ wcel 2202   ≠ wne 2402  ‘cfv 5326  0_gc0g 13338  1_rcur 13971  Ringcrg 14008  NzRingcnzr 14192

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rab 2519  df-in 3206  df-ss 3213  df-nzr 14193

This theorem is referenced by:  nzrunit  14201  lringring  14207  rrgnz  14281  domnring  14284