Theorem lringnzr 20793

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 20791	. . 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 4007	. 2 ⊢ LRing ⊆ NzRing
3	2	sseli 3904	1 ⊢ (𝑅 ∈ LRing → 𝑅 ∈ NzRing)

This proof depends on syntax axioms:   → wi 4   ∨ wo 861   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ‘cfv 6538  (class class class)co 7420  Basecbs 17387  +_gcplusg 17428  1_rcur 20407  Unitcui 20585  NzRingcnzr 20762  LRingclring 20790

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-ss 3896  df-lring 20791

This theorem is used by:  lringring  20794  lringnz  20795  dflring2  34025  dflringlem2  34027