Theorem nzrring 14278

Description: A nonzero ring is a ring. (Contributed by Stefan O'Rear, 24-Feb-2015.) (Proof shortened by SN, 23-Feb-2025.)

Assertion

Ref	Expression
nzrring	|- ( R e. NzRing -> R e. Ring )

Proof of Theorem nzrring

Dummy variable r is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-nzr 14275	. . 3 |- NzRing = { r e. Ring | ( 1r ` r ) =/= ( 0g ` r ) }
2	1	ssrab3 3314	. 2 |- NzRing C_ Ring
3	2	sseli 3224	1 |- ( R e. NzRing -> R e. Ring )

Syntax hints:   -> wi 4   e. wcel 2202   =/= wne 2403   ` cfv 5333   0gc0g 13419   1rcur 14053   Ringcrg 14090  NzRingcnzr 14274

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rab 2520  df-in 3207  df-ss 3214  df-nzr 14275

This theorem is referenced by:  nzrunit  14283  lringring  14289  rrgnz  14364  domnring  14367