Theorem supmaxti 7203

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 533	. . 3 |- ( ( ph /\ ( y e. A /\ y R C ) ) -> y R C )
6		breq2 4092	. . . 4 |- ( x = C -> ( y R x <-> y R C ) )
7	6	rspcev 2910	. . 3 |- ( ( C e. B /\ y R C ) -> E. x e. B y R x )
8	4, 5, 7	syl2an2r 599	. 2 |- ( ( ph /\ ( y e. A /\ y R C ) ) -> E. x e. B y R x )
9	1, 2, 3, 8	eqsuptid 7196	1 |- ( ph -> sup ( B , A , R ) = C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   E.wrex 2511   class class class wbr 4088   supcsup 7181

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-un 3204  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-iota 5286  df-riota 5971  df-sup 7183

This theorem is referenced by:  supsnti  7204  sup3exmid  9137  maxleim  11766  xrmaxleim  11805  supfz  16678