Theorem isidom 14225

Description: An integral domain is a commutative domain. (Contributed by Mario Carneiro, 17-Jun-2015.)

Assertion

Ref	Expression
isidom	|- ( R e. IDomn <-> ( R e. CRing /\ R e. Domn ) )

Proof of Theorem isidom

Step	Hyp	Ref	Expression
1		df-idom 14209	. 2 |- IDomn = ( CRing i^i Domn )
2	1	elin2 3392	1 |- ( R e. IDomn <-> ( R e. CRing /\ R e. Domn ) )

Syntax hints:   /\ wa 104   <-> wb 105   e. wcel 2200   CRingccrg 13946  Domncdomn 14205  IDomncidom 14206

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-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-in 3203  df-idom 14209

This theorem is referenced by:  znidom  14606  znidomb  14607