Theorem lringnzr 14070

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 14068	. . 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 3287	. 2 |- LRing C_ NzRing
3	2	sseli 3197	1 |- ( R e. LRing -> R e. NzRing )

Syntax hints:   -> wi 4   \/ wo 710   = wceq 1373   e. wcel 2178   A.wral 2486   ` cfv 5290  (class class class)co 5967   Basecbs 12947   +g cplusg 13024   1rcur 13836  Unitcui 13964  NzRingcnzr 14056  LRingclring 14067

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rab 2495  df-in 3180  df-ss 3187  df-lring 14068

This theorem is referenced by:  lringring  14071  lringnz  14072