Theorem nzrring 13739

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 13736	. . 3 |- NzRing = { r e. Ring | ( 1r ` r ) =/= ( 0g ` r ) }
2	1	ssrab3 3269	. 2 |- NzRing C_ Ring
3	2	sseli 3179	1 |- ( R e. NzRing -> R e. Ring )

Syntax hints:   -> wi 4   e. wcel 2167   =/= wne 2367   ` cfv 5258   0gc0g 12927   1rcur 13515   Ringcrg 13552  NzRingcnzr 13735

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-nzr 13736

This theorem is referenced by:  nzrunit  13744  lringring  13750  rrgnz  13824  domnring  13827