Theorem lringnzr 13265

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

Assertion

Ref	Expression
lringnzr	|- ( R e. LRing -> R e. 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 13263	. . 3 |- LRing = { r e. NzRing | A. x e. ( Base ` r ) A. y e. ( Base ` r ) ( ( x ( +g ` r ) y ) = ( 1r ` r ) -> ( x e. (Unit ` r ) \/ y e. (Unit ` r ) ) ) }
2	1	ssrab3 3241	. 2 |- LRing C_ NzRing
3	2	sseli 3151	1 |- ( R e. LRing -> R e. NzRing )

Syntax hints:   -> wi 4   \/ wo 708   = wceq 1353   e. wcel 2148   A.wral 2455   ` cfv 5215  (class class class)co 5872   Basecbs 12454   +g cplusg 12528   1rcur 13073  Unitcui 13187  NzRingcnzr 13254  LRingclring 13262

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rab 2464  df-in 3135  df-ss 3142  df-lring 13263

This theorem is referenced by:  lringring  13266  lringnz  13267