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Theorem maxidlnr 26770
Description: A maximal ideal is proper. (Contributed by Jeff Madsen, 16-Jun-2011.)
Hypotheses
Ref Expression
maxidlnr.1  |-  G  =  ( 1st `  R
)
maxidlnr.2  |-  X  =  ran  G
Assertion
Ref Expression
maxidlnr  |-  ( ( R  e.  RingOps  /\  M  e.  ( MaxIdl `  R )
)  ->  M  =/=  X )

Proof of Theorem maxidlnr
Dummy variable  j is distinct from all other variables.
StepHypRef Expression
1 maxidlnr.1 . . . 4  |-  G  =  ( 1st `  R
)
2 maxidlnr.2 . . . 4  |-  X  =  ran  G
31, 2ismaxidl 26768 . . 3  |-  ( R  e.  RingOps  ->  ( M  e.  ( MaxIdl `  R )  <->  ( M  e.  ( Idl `  R )  /\  M  =/=  X  /\  A. j  e.  ( Idl `  R
) ( M  C_  j  ->  ( j  =  M  \/  j  =  X ) ) ) ) )
43biimpa 470 . 2  |-  ( ( R  e.  RingOps  /\  M  e.  ( MaxIdl `  R )
)  ->  ( M  e.  ( Idl `  R
)  /\  M  =/=  X  /\  A. j  e.  ( Idl `  R
) ( M  C_  j  ->  ( j  =  M  \/  j  =  X ) ) ) )
54simp2d 968 1  |-  ( ( R  e.  RingOps  /\  M  e.  ( MaxIdl `  R )
)  ->  M  =/=  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556    C_ wss 3165   ran crn 4706   ` cfv 5271   1stc1st 6136   RingOpscrngo 21058   Idlcidl 26735   MaxIdlcmaxidl 26737
This theorem is referenced by:  maxidln1  26772
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fv 5279  df-maxidl 26740
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