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Theorem ismxl2 25267
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 25265 . . 3  |-  ( R  e. PresetRel  ->  ( mxl `  R
)  =  { a  e.  X  |  A. b  e.  X  (
a R b  -> 
a  =  b ) } )
32eleq2d 2350 . 2  |-  ( R  e. PresetRel  ->  ( A  e.  ( mxl `  R
)  <->  A  e.  { a  e.  X  |  A. b  e.  X  (
a R b  -> 
a  =  b ) } ) )
4 breq1 4026 . . . . 5  |-  ( a  =  A  ->  (
a R b  <->  A R
b ) )
5 eqeq1 2289 . . . . 5  |-  ( a  =  A  ->  (
a  =  b  <->  A  =  b ) )
64, 5imbi12d 311 . . . 4  |-  ( a  =  A  ->  (
( a R b  ->  a  =  b )  <->  ( A R b  ->  A  =  b ) ) )
76ralbidv 2563 . . 3  |-  ( a  =  A  ->  ( A. b  e.  X  ( a R b  ->  a  =  b )  <->  A. b  e.  X  ( A R b  ->  A  =  b )
) )
87elrab 2923 . 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 1623    e. wcel 1684   A.wral 2543   {crab 2547   class class class wbr 4023   dom cdm 4689   ` cfv 5255  PresetRelcpresetrel 25215   mxlcmxl 25216
This theorem is referenced by:  geme2  25275
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-iota 5219  df-fun 5257  df-fv 5263  df-prs 25223  df-mxl 25246
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