Theorem lringnzr 14500

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 14498	. . 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 3334	. 2 ⊢ LRing ⊆ NzRing
3	2	sseli 3244	1 ⊢ (𝑅 ∈ LRing → 𝑅 ∈ NzRing)

This proof depends on syntax axioms:   → wi 4   ∨ wo 720   = wceq 1402   ∈ wcel 2209  ∀wral 2528  ‘cfv 5377  (class class class)co 6085  Basecbs 13352  +_gcplusg 13431  1_rcur 14262  Unitcui 14393  NzRingcnzr 14486  LRingclring 14497

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rab 2537  df-in 3226  df-ss 3233  df-lring 14498

This theorem is used by:  lringring  14501  lringnz  14502  drngunz  14618