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Theorem defge3 25271
Description: The greatest element of a poset is an element, when it exists, that is greater than the other elements of the poset. Use the idiom  ( ge `  R )  e.  X when you mean the greatest element of  X exists. (Contributed by FL, 30-Dec-2011.)
Hypothesis
Ref Expression
defge3.1  |-  X  =  dom  R
Assertion
Ref Expression
defge3  |-  ( ( R  e.  PosetRel  /\  ( ge `  R )  e.  X )  ->  A. x  e.  X  x R
( ge `  R
) )
Distinct variable groups:    x, R    x, X

Proof of Theorem defge3
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 defge3.1 . . . 4  |-  X  =  dom  R
21gepsup 25250 . . 3  |-  ( R  e.  PosetRel  ->  ( ge `  R )  =  ( R  sup w  X
) )
3 simpl 443 . . . . . 6  |-  ( ( R  e.  PosetRel  /\  ( R  sup w  X )  e.  X )  ->  R  e.  PosetRel )
4 dmexg 4939 . . . . . . . 8  |-  ( R  e.  PosetRel  ->  dom  R  e.  _V )
51, 4syl5eqel 2367 . . . . . . 7  |-  ( R  e.  PosetRel  ->  X  e.  _V )
65adantr 451 . . . . . 6  |-  ( ( R  e.  PosetRel  /\  ( R  sup w  X )  e.  X )  ->  X  e.  _V )
7 simpr 447 . . . . . 6  |-  ( ( R  e.  PosetRel  /\  ( R  sup w  X )  e.  X )  -> 
( R  sup w  X )  e.  X
)
81supdef 25262 . . . . . 6  |-  ( ( R  e.  PosetRel  /\  X  e.  _V  /\  ( R  sup w  X )  e.  X )  -> 
( A. x  e.  X  x R ( R  sup w  X
)  /\  A. x  e.  X  ( A. y  e.  X  y R x  ->  ( R  sup w  X ) R x ) ) )
93, 6, 7, 8syl3anc 1182 . . . . 5  |-  ( ( R  e.  PosetRel  /\  ( R  sup w  X )  e.  X )  -> 
( A. x  e.  X  x R ( R  sup w  X
)  /\  A. x  e.  X  ( A. y  e.  X  y R x  ->  ( R  sup w  X ) R x ) ) )
109simpld 445 . . . 4  |-  ( ( R  e.  PosetRel  /\  ( R  sup w  X )  e.  X )  ->  A. x  e.  X  x R ( R  sup w  X ) )
11 eleq1 2343 . . . . . 6  |-  ( ( ge `  R )  =  ( R  sup w  X )  ->  (
( ge `  R
)  e.  X  <->  ( R  sup w  X )  e.  X ) )
1211anbi2d 684 . . . . 5  |-  ( ( ge `  R )  =  ( R  sup w  X )  ->  (
( R  e.  PosetRel  /\  ( ge `  R )  e.  X )  <->  ( R  e. 
PosetRel  /\  ( R  sup w  X )  e.  X
) ) )
13 breq2 4027 . . . . . 6  |-  ( ( ge `  R )  =  ( R  sup w  X )  ->  (
x R ( ge
`  R )  <->  x R
( R  sup w  X ) ) )
1413ralbidv 2563 . . . . 5  |-  ( ( ge `  R )  =  ( R  sup w  X )  ->  ( A. x  e.  X  x R ( ge `  R )  <->  A. x  e.  X  x R
( R  sup w  X ) ) )
1512, 14imbi12d 311 . . . 4  |-  ( ( ge `  R )  =  ( R  sup w  X )  ->  (
( ( R  e.  PosetRel 
/\  ( ge `  R )  e.  X
)  ->  A. x  e.  X  x R
( ge `  R
) )  <->  ( ( R  e.  PosetRel  /\  ( R  sup w  X )  e.  X )  ->  A. x  e.  X  x R ( R  sup w  X ) ) ) )
1610, 15mpbiri 224 . . 3  |-  ( ( ge `  R )  =  ( R  sup w  X )  ->  (
( R  e.  PosetRel  /\  ( ge `  R )  e.  X )  ->  A. x  e.  X  x R ( ge `  R ) ) )
172, 16syl 15 . 2  |-  ( R  e.  PosetRel  ->  ( ( R  e.  PosetRel  /\  ( ge `  R )  e.  X
)  ->  A. x  e.  X  x R
( ge `  R
) ) )
1817anabsi5 790 1  |-  ( ( R  e.  PosetRel  /\  ( ge `  R )  e.  X )  ->  A. x  e.  X  x R
( ge `  R
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788   class class class wbr 4023   dom cdm 4689   ` cfv 5255  (class class class)co 5858   PosetRelcps 14301    sup
w cspw 14303   gecge 25220
This theorem is referenced by:  defse3  25272
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-pow 4188  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-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  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-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-ov 5861  df-oprab 5862  df-mpt2 5863  df-undef 6298  df-riota 6304  df-ps 14306  df-spw 14308  df-ge 25248
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