Theorem supmaxti 6950

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 Jim Kingdon, 24-Nov-2021.)

Hypotheses

Ref	Expression
supmaxti.ti	|- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )
supmaxti.2	|- ( ph -> C e. A )
supmaxti.3	|- ( ph -> C e. B )
supmaxti.4	|- ( ( ph /\ y e. B ) -> -. C R y )

Assertion

Ref	Expression
supmaxti	|- ( ph -> sup ( B , A , R ) = C )

Distinct variable groups:   u, A, v, y   u, B, v, y   u, C, v, y   u, R, v, y   ph, u, v, y

Proof of Theorem supmaxti

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		supmaxti.ti	. 2 |- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )
2		supmaxti.2	. 2 |- ( ph -> C e. A )
3		supmaxti.4	. 2 |- ( ( ph /\ y e. B ) -> -. C R y )
4		supmaxti.3	. . 3 |- ( ph -> C e. B )
5		simprr 522	. . 3 |- ( ( ph /\ ( y e. A /\ y R C ) ) -> y R C )
6		breq2 3971	. . . 4 |- ( x = C -> ( y R x <-> y R C ) )
7	6	rspcev 2816	. . 3 |- ( ( C e. B /\ y R C ) -> E. x e. B y R x )
8	4, 5, 7	syl2an2r 585	. 2 |- ( ( ph /\ ( y e. A /\ y R C ) ) -> E. x e. B y R x )
9	1, 2, 3, 8	eqsuptid 6943	1 |- ( ph -> sup ( B , A , R ) = C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1335   e. wcel 2128   E.wrex 2436   class class class wbr 3967   supcsup 6928

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-un 3106  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-br 3968  df-iota 5137  df-riota 5782  df-sup 6930

This theorem is referenced by:  supsnti  6951  sup3exmid  8833  maxleim  11116  xrmaxleim  11152  supfz  13710