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Theorem supmax 7470
Description: The greatest element of a set is its supremum. Note that the converse is not true; the supremum might not be an element of the set considered. (Contributed by Jeff Hoffman, 17-Jun-2008.)
Hypotheses
Ref Expression
supmax.1  |-  ( ph  ->  R  Or  A )
supmax.2  |-  ( ph  ->  C  e.  A )
supmax.3  |-  ( ph  ->  C  e.  B )
supmax.4  |-  ( (
ph  /\  y  e.  B )  ->  -.  C R y )
Assertion
Ref Expression
supmax  |-  ( ph  ->  sup ( B ,  A ,  R )  =  C )
Distinct variable groups:    y, A    y, B    y, C    y, R    ph, y

Proof of Theorem supmax
Dummy variables  x  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 supmax.3 . . 3  |-  ( ph  ->  C  e.  B )
2 supmax.1 . . . 4  |-  ( ph  ->  R  Or  A )
3 supmax.2 . . . . 5  |-  ( ph  ->  C  e.  A )
4 supmax.4 . . . . . 6  |-  ( (
ph  /\  y  e.  B )  ->  -.  C R y )
54ralrimiva 2789 . . . . 5  |-  ( ph  ->  A. y  e.  B  -.  C R y )
6 supmaxlem 7469 . . . . 5  |-  ( ( C  e.  A  /\  C  e.  B  /\  A. y  e.  B  -.  C R y )  ->  E. x  e.  A  ( A. z  e.  B  -.  x R z  /\  A. z  e.  A  ( z R x  ->  E. y  e.  B  z R y ) ) )
73, 1, 5, 6syl3anc 1184 . . . 4  |-  ( ph  ->  E. x  e.  A  ( A. z  e.  B  -.  x R z  /\  A. z  e.  A  ( z R x  ->  E. y  e.  B  z R y ) ) )
82, 7supub 7464 . . 3  |-  ( ph  ->  ( C  e.  B  ->  -.  sup ( B ,  A ,  R
) R C ) )
91, 8mpd 15 . 2  |-  ( ph  ->  -.  sup ( B ,  A ,  R
) R C )
102, 7supnub 7467 . . 3  |-  ( ph  ->  ( ( C  e.  A  /\  A. y  e.  B  -.  C R y )  ->  -.  C R sup ( B ,  A ,  R ) ) )
113, 5, 10mp2and 661 . 2  |-  ( ph  ->  -.  C R sup ( B ,  A ,  R ) )
122, 7supcl 7463 . . 3  |-  ( ph  ->  sup ( B ,  A ,  R )  e.  A )
13 sotrieq2 4531 . . 3  |-  ( ( R  Or  A  /\  ( sup ( B ,  A ,  R )  e.  A  /\  C  e.  A ) )  -> 
( sup ( B ,  A ,  R
)  =  C  <->  ( -.  sup ( B ,  A ,  R ) R C  /\  -.  C R sup ( B ,  A ,  R )
) ) )
142, 12, 3, 13syl12anc 1182 . 2  |-  ( ph  ->  ( sup ( B ,  A ,  R
)  =  C  <->  ( -.  sup ( B ,  A ,  R ) R C  /\  -.  C R sup ( B ,  A ,  R )
) ) )
159, 11, 14mpbir2and 889 1  |-  ( ph  ->  sup ( B ,  A ,  R )  =  C )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   E.wrex 2706   class class class wbr 4212    Or wor 4502   supcsup 7445
This theorem is referenced by:  suppr  7473  lbinfm  9961  ramcl2lem  13377  gsumesum  24451  ballotlemirc  24789  supfz  25199  inffz  25200  mblfinlem2  26244
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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-po 4503  df-so 4504  df-iota 5418  df-riota 6549  df-sup 7446
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