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Theorem spweu 14352
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 14351 . . . 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 2766 . . 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 1696   A.wral 2556   E.wrex 2557   E!wreu 2558   E*wrmo 2559   class class class wbr 4039   PosetRelcps 14317
This theorem is referenced by:  spwex  14354  spwpr4  14356  supexr  25734
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-res 4717  df-ps 14322
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