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Theorem spweu 14651
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 14650 . . . 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 2913 . . 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 1725   A.wral 2697   E.wrex 2698   E!wreu 2699   E*wrmo 2700   class class class wbr 4204   PosetRelcps 14616
This theorem is referenced by:  spwex  14653  spwpr4  14655
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-res 4882  df-ps 14621
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