Theorem lringnzr 20569

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 20567	. . 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 4105	. 2 ⊢ LRing ⊆ NzRing
3	2	sseli 4004	1 ⊢ (𝑅 ∈ LRing → 𝑅 ∈ NzRing)

Syntax hints:   → wi 4   ∨ wo 846   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ‘cfv 6575  (class class class)co 7450  Basecbs 17260  +_gcplusg 17313  1_rcur 20210  Unitcui 20383  NzRingcnzr 20540  LRingclring 20566

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-ss 3993  df-lring 20567

This theorem is referenced by:  lringring  20570  lringnz  20571