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Theorem ismxl2 25370
Description: The predicate " A is a maximal element of the preset  R " . (Contributed by FL, 22-May-2011.)
Hypothesis
Ref Expression
ismxl2.1  |-  X  =  dom  R
Assertion
Ref Expression
ismxl2  |-  ( R  e. PresetRel  ->  ( A  e.  ( mxl `  R
)  <->  ( A  e.  X  /\  A. b  e.  X  ( A R b  ->  A  =  b ) ) ) )
Distinct variable groups:    A, b    R, b    X, b

Proof of Theorem ismxl2
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 ismxl2.1 . . . 4  |-  X  =  dom  R
21mxlelt2 25368 . . 3  |-  ( R  e. PresetRel  ->  ( mxl `  R
)  =  { a  e.  X  |  A. b  e.  X  (
a R b  -> 
a  =  b ) } )
32eleq2d 2363 . 2  |-  ( R  e. PresetRel  ->  ( A  e.  ( mxl `  R
)  <->  A  e.  { a  e.  X  |  A. b  e.  X  (
a R b  -> 
a  =  b ) } ) )
4 breq1 4042 . . . . 5  |-  ( a  =  A  ->  (
a R b  <->  A R
b ) )
5 eqeq1 2302 . . . . 5  |-  ( a  =  A  ->  (
a  =  b  <->  A  =  b ) )
64, 5imbi12d 311 . . . 4  |-  ( a  =  A  ->  (
( a R b  ->  a  =  b )  <->  ( A R b  ->  A  =  b ) ) )
76ralbidv 2576 . . 3  |-  ( a  =  A  ->  ( A. b  e.  X  ( a R b  ->  a  =  b )  <->  A. b  e.  X  ( A R b  ->  A  =  b )
) )
87elrab 2936 . 2  |-  ( A  e.  { a  e.  X  |  A. b  e.  X  ( a R b  ->  a  =  b ) }  <-> 
( A  e.  X  /\  A. b  e.  X  ( A R b  ->  A  =  b )
) )
93, 8syl6bb 252 1  |-  ( R  e. PresetRel  ->  ( A  e.  ( mxl `  R
)  <->  ( A  e.  X  /\  A. b  e.  X  ( A R b  ->  A  =  b ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   {crab 2560   class class class wbr 4039   dom cdm 4705   ` cfv 5271  PresetRelcpresetrel 25318   mxlcmxl 25319
This theorem is referenced by:  geme2  25378
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-13 1698  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  ax-un 4528
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-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  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-res 4717  df-iota 5235  df-fun 5273  df-fv 5279  df-prs 25326  df-mxl 25349
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