Theorem lringnzr 14210

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 14208	. . 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 3313	. 2 |- LRing C_ NzRing
3	2	sseli 3223	1 |- ( R e. LRing -> R e. NzRing )

Syntax hints:   -> wi 4   \/ wo 715   = wceq 1397   e. wcel 2202   A.wral 2510   ` cfv 5326  (class class class)co 6018   Basecbs 13084   +g cplusg 13162   1rcur 13975  Unitcui 14103  NzRingcnzr 14196  LRingclring 14207

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rab 2519  df-in 3206  df-ss 3213  df-lring 14208

This theorem is referenced by:  lringring  14211  lringnz  14212