Theorem lringnzr 20509

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 20507	. . 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 4062	. 2 ⊢ LRing ⊆ NzRing
3	2	sseli 3959	1 ⊢ (𝑅 ∈ LRing → 𝑅 ∈ NzRing)

Syntax hints:   → wi 4   ∨ wo 847   = wceq 1539   ∈ wcel 2107  ∀wral 3050  ‘cfv 6541  (class class class)co 7413  Basecbs 17229  +_gcplusg 17273  1_rcur 20146  Unitcui 20323  NzRingcnzr 20480  LRingclring 20506

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-rab 3420  df-ss 3948  df-lring 20507

This theorem is referenced by:  lringring  20510  lringnz  20511