Theorem issdrng 17179

Description: The predicate "is a *-division ring."

Assertion

Ref	Expression
issdrng	|- (R e. DivRing* <-> (R e. DivRing /\ R e. Ring*))

Proof of Theorem issdrng

Step	Hyp	Ref	Expression
1		df-sdivrng 17178	. . 3 |- DivRing* = (DivRing i^i Ring*)
2	1	eleq2i 1961	. 2 |- (R e. DivRing* <-> R e. (DivRing i^i Ring*))
3		elin 2786	. 2 |- (R e. (DivRing i^i Ring*) <-> (R e. DivRing /\ R e. Ring*))
4	2, 3	bitri 190	1 |- (R e. DivRing* <-> (R e. DivRing /\ R e. Ring*))

Syntax hints:   <-> wb 163   /\ wa 240   e. wcel 1300   i^i cin 2592  DivRingcdrng 9491  Ring*csr 17173  DivRing*csd 17177

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-sdivrng 17178

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