Theorem iscring 16145

Description: The predicate "is a commutative ring".

Assertion

Ref	Expression
iscring	|- (R e. CRing <-> (R e. Ring /\ R e. Com2))

Proof of Theorem iscring

Step	Hyp	Ref	Expression
1		df-cring 16144	. . 3 |- CRing = (Ring i^i Com2)
2	1	eleq2i 1961	. 2 |- (R e. CRing <-> R e. (Ring i^i Com2))
3		elin 2786	. 2 |- (R e. (Ring i^i Com2) <-> (R e. Ring /\ R e. Com2))
4	2, 3	bitri 190	1 |- (R e. CRing <-> (R e. Ring /\ R e. Com2))

Syntax hints:   <-> wb 163   /\ wa 240   e. wcel 1300   i^i cin 2592  Ringcring 9463  Com2ccm2 10394  CRingccring 16143

This theorem is referenced by:  iscring2 16146  crngrng 16148  fldcrng 16152  isfld2 16153  isdmn2 16203

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-in 2603  df-cring 16144

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