Theorem nzrring 13978

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 13975	. . 3 |- NzRing = { r e. Ring | ( 1r ` r ) =/= ( 0g ` r ) }
2	1	ssrab3 3279	. 2 |- NzRing C_ Ring
3	2	sseli 3189	1 |- ( R e. NzRing -> R e. Ring )

Syntax hints:   -> wi 4   e. wcel 2176   =/= wne 2376   ` cfv 5272   0gc0g 13121   1rcur 13754   Ringcrg 13791  NzRingcnzr 13974

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rab 2493  df-in 3172  df-ss 3179  df-nzr 13975

This theorem is referenced by:  nzrunit  13983  lringring  13989  rrgnz  14063  domnring  14066