Theorem isfld 14597

Description: A field is a commutative division ring. (Contributed by Mario Carneiro, 17-Jun-2015.)

Assertion

Ref	Expression
isfld	|- ( R e. Field <-> ( R e. DivRing /\ R e. CRing ) )

Proof of Theorem isfld

Step	Hyp	Ref	Expression
1		df-field 14588	. 2 |- Field = ( DivRing i^i CRing )
2	1	elin2 3417	1 |- ( R e. Field <-> ( R e. DivRing /\ R e. CRing ) )

Syntax hints:   /\ wa 104   <-> wb 105   e. wcel 2209   CRingccrg 14284   DivRingcdr 14585  Fieldcfield 14586

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-field 14588

This theorem is referenced by:  flddrngd  14598  fldcrngd  14599