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Theorem supmax 7216
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 2626 . . . . 5  |-  ( ph  ->  A. y  e.  B  -.  C R y )
6 supmaxlem 7215 . . . . 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 1182 . . . 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 7210 . . 3  |-  ( ph  ->  ( C  e.  B  ->  -.  sup ( B ,  A ,  R
) R C ) )
91, 8mpd 14 . 2  |-  ( ph  ->  -.  sup ( B ,  A ,  R
) R C )
102, 7supnub 7213 . . 3  |-  ( ph  ->  ( ( C  e.  A  /\  A. y  e.  B  -.  C R y )  ->  -.  C R sup ( B ,  A ,  R ) ) )
113, 5, 10mp2and 660 . 2  |-  ( ph  ->  -.  C R sup ( B ,  A ,  R ) )
122, 7supcl 7209 . . 3  |-  ( ph  ->  sup ( B ,  A ,  R )  e.  A )
13 sotrieq2 4342 . . 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 1180 . 2  |-  ( ph  ->  ( sup ( B ,  A ,  R
)  =  C  <->  ( -.  sup ( B ,  A ,  R ) R C  /\  -.  C R sup ( B ,  A ,  R )
) ) )
159, 11, 14mpbir2and 888 1  |-  ( ph  ->  sup ( B ,  A ,  R )  =  C )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   class class class wbr 4023    Or wor 4313   supcsup 7193
This theorem is referenced by:  ballotlemirc  23090  supfz  24094  inffz  24095
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  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-po 4314  df-so 4315  df-iota 5219  df-riota 6304  df-sup 7194
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