Theorem lringnzr 14206

Description: A local ring is a nonzero ring. (Contributed by SN, 23-Feb-2025.)

Assertion

Ref	Expression
lringnzr	⊢ (𝑅 ∈ LRing → 𝑅 ∈ NzRing)

Proof of Theorem lringnzr

Dummy variables 𝑟 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-lring 14204	. . 3 ⊢ LRing = {𝑟 ∈ NzRing ∣ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)((𝑥(+g‘𝑟)𝑦) = (1r‘𝑟) → (𝑥 ∈ (Unit‘𝑟) ∨ 𝑦 ∈ (Unit‘𝑟)))}
2	1	ssrab3 3313	. 2 ⊢ LRing ⊆ NzRing
3	2	sseli 3223	1 ⊢ (𝑅 ∈ LRing → 𝑅 ∈ NzRing)

Syntax hints:   → wi 4   ∨ wo 715   = wceq 1397   ∈ wcel 2202  ∀wral 2510  ‘cfv 5326  (class class class)co 6017  Basecbs 13081  +_gcplusg 13159  1_rcur 13971  Unitcui 14099  NzRingcnzr 14192  LRingclring 14203

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-lring 14204

This theorem is referenced by:  lringring  14207  lringnz  14208