Theorem lringnzr 13955

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 13953	. . 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 3279	. 2 |- LRing C_ NzRing
3	2	sseli 3189	1 |- ( R e. LRing -> R e. NzRing )

Syntax hints:   -> wi 4   \/ wo 710   = wceq 1373   e. wcel 2176   A.wral 2484   ` cfv 5271  (class class class)co 5944   Basecbs 12832   +g cplusg 12909   1rcur 13721  Unitcui 13849  NzRingcnzr 13941  LRingclring 13952

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rab 2493  df-in 3172  df-ss 3179  df-lring 13953

This theorem is referenced by:  lringring  13956  lringnz  13957