Theorem isidom 14108

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 14092	. 2 |- IDomn = ( CRing i^i Domn )
2	1	elin2 3365	1 |- ( R e. IDomn <-> ( R e. CRing /\ R e. Domn ) )

Syntax hints:   /\ wa 104   <-> wb 105   e. wcel 2177   CRingccrg 13829  Domncdomn 14088  IDomncidom 14089

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-in 3176  df-idom 14092

This theorem is referenced by:  znidom  14489  znidomb  14490