Theorem lringnzr 13749

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 13747	. . 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 3269	. 2 |- LRing C_ NzRing
3	2	sseli 3179	1 |- ( R e. LRing -> R e. NzRing )

Syntax hints:   -> wi 4   \/ wo 709   = wceq 1364   e. wcel 2167   A.wral 2475   ` cfv 5258  (class class class)co 5922   Basecbs 12678   +g cplusg 12755   1rcur 13515  Unitcui 13643  NzRingcnzr 13735  LRingclring 13746

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rab 2484  df-in 3163  df-ss 3170  df-lring 13747

This theorem is referenced by:  lringring  13750  lringnz  13751