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Theorem spweu 14588
Description: A supremum is unique. (Contributed by NM, 15-May-2008.)
Hypothesis
Ref Expression
spwmo.1  |-  ( ph  <->  ( A. y  e.  A  y R x  /\  A. y  e.  X  ( A. z  e.  A  z R y  ->  x R y ) ) )
Assertion
Ref Expression
spweu  |-  ( ( R  e.  PosetRel  /\  E. x  e.  X  ph )  ->  E! x  e.  X  ph )
Distinct variable groups:    x, y,
z, A    x, R, y, z    x, X, y
Allowed substitution hints:    ph( x, y, z)    X( z)

Proof of Theorem spweu
StepHypRef Expression
1 spwmo.1 . . . . 5  |-  ( ph  <->  ( A. y  e.  A  y R x  /\  A. y  e.  X  ( A. z  e.  A  z R y  ->  x R y ) ) )
21spwmo 14587 . . . 4  |-  ( R  e.  PosetRel  ->  E* x  e.  X ph )
32anim2i 553 . . 3  |-  ( ( E. x  e.  X  ph 
/\  R  e.  PosetRel )  ->  ( E. x  e.  X  ph  /\  E* x  e.  X ph ) )
4 reu5 2866 . . 3  |-  ( E! x  e.  X  ph  <->  ( E. x  e.  X  ph 
/\  E* x  e.  X ph ) )
53, 4sylibr 204 . 2  |-  ( ( E. x  e.  X  ph 
/\  R  e.  PosetRel )  ->  E! x  e.  X  ph )
65ancoms 440 1  |-  ( ( R  e.  PosetRel  /\  E. x  e.  X  ph )  ->  E! x  e.  X  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1717   A.wral 2651   E.wrex 2652   E!wreu 2653   E*wrmo 2654   class class class wbr 4155   PosetRelcps 14553
This theorem is referenced by:  spwex  14590  spwpr4  14592
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pr 4346
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-res 4832  df-ps 14558
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