Theorem nzrring 13682

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 13679	. . 3 |- NzRing = { r e. Ring | ( 1r ` r ) =/= ( 0g ` r ) }
2	1	ssrab3 3266	. 2 |- NzRing C_ Ring
3	2	sseli 3176	1 |- ( R e. NzRing -> R e. Ring )

Syntax hints:   -> wi 4   e. wcel 2164   =/= wne 2364   ` cfv 5255   0gc0g 12870   1rcur 13458   Ringcrg 13495  NzRingcnzr 13678

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rab 2481  df-in 3160  df-ss 3167  df-nzr 13679

This theorem is referenced by:  nzrunit  13687  lringring  13693  rrgnz  13767  domnring  13770