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Theorem spweu 14336
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 14335 . . . 4  |-  ( R  e.  PosetRel  ->  E* x  e.  X ph )
32anim2i 552 . . 3  |-  ( ( E. x  e.  X  ph 
/\  R  e.  PosetRel )  ->  ( E. x  e.  X  ph  /\  E* x  e.  X ph ) )
4 reu5 2753 . . 3  |-  ( E! x  e.  X  ph  <->  ( E. x  e.  X  ph 
/\  E* x  e.  X ph ) )
53, 4sylibr 203 . 2  |-  ( ( E. x  e.  X  ph 
/\  R  e.  PosetRel )  ->  E! x  e.  X  ph )
65ancoms 439 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 176    /\ wa 358    e. wcel 1684   A.wral 2543   E.wrex 2544   E!wreu 2545   E*wrmo 2546   class class class wbr 4023   PosetRelcps 14301
This theorem is referenced by:  spwex  14338  spwpr4  14340  supexr  25631
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-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
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-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-res 4701  df-ps 14306
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