Theorem nzrring 14147

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 14144	. . 3 |- NzRing = { r e. Ring | ( 1r ` r ) =/= ( 0g ` r ) }
2	1	ssrab3 3310	. 2 |- NzRing C_ Ring
3	2	sseli 3220	1 |- ( R e. NzRing -> R e. Ring )

Syntax hints:   -> wi 4   e. wcel 2200   =/= wne 2400   ` cfv 5318   0gc0g 13289   1rcur 13922   Ringcrg 13959  NzRingcnzr 14143

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rab 2517  df-in 3203  df-ss 3210  df-nzr 14144

This theorem is referenced by:  nzrunit  14152  lringring  14158  rrgnz  14232  domnring  14235