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Theorem supmax 8012
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.) (Proof shortened by OpenAI, 30-Mar-2020.)
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 variable  z is distinct from all other variables.
StepHypRef Expression
1 supmax.1 . 2  |-  ( ph  ->  R  Or  A )
2 supmax.2 . 2  |-  ( ph  ->  C  e.  A )
3 supmax.4 . 2  |-  ( (
ph  /\  y  e.  B )  ->  -.  C R y )
4 supmax.3 . . . 4  |-  ( ph  ->  C  e.  B )
54adantr 471 . . 3  |-  ( (
ph  /\  ( y  e.  A  /\  y R C ) )  ->  C  e.  B )
6 simprr 771 . . 3  |-  ( (
ph  /\  ( y  e.  A  /\  y R C ) )  -> 
y R C )
7 breq2 4422 . . . 4  |-  ( z  =  C  ->  (
y R z  <->  y R C ) )
87rspcev 3162 . . 3  |-  ( ( C  e.  B  /\  y R C )  ->  E. z  e.  B  y R z )
95, 6, 8syl2anc 671 . 2  |-  ( (
ph  /\  ( y  e.  A  /\  y R C ) )  ->  E. z  e.  B  y R z )
101, 2, 3, 9eqsupd 8002 1  |-  ( ph  ->  sup ( B ,  A ,  R )  =  C )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 375    = wceq 1455    e. wcel 1898   E.wrex 2750   class class class wbr 4418    Or wor 4776   supcsup 7985
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-br 4419  df-po 4777  df-so 4778  df-iota 5569  df-riota 6282  df-sup 7987
This theorem is referenced by:  suppr  8018  lbinf  10592  lbinfmOLD  10593  ramcl2lemOLD  15018  gsumesum  28931  oms0OLD  29179  ballotlemircOLD  29452  supfz  30412  inffz  30413  mblfinlem2  32024
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