Theorem supmaxti 7002

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 531	. . 3 |- ( ( ph /\ ( y e. A /\ y R C ) ) -> y R C )
6		breq2 4007	. . . 4 |- ( x = C -> ( y R x <-> y R C ) )
7	6	rspcev 2841	. . 3 |- ( ( C e. B /\ y R C ) -> E. x e. B y R x )
8	4, 5, 7	syl2an2r 595	. 2 |- ( ( ph /\ ( y e. A /\ y R C ) ) -> E. x e. B y R x )
9	1, 2, 3, 8	eqsuptid 6995	1 |- ( ph -> sup ( B , A , R ) = C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2148   E.wrex 2456   class class class wbr 4003   supcsup 6980

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-un 3133  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-iota 5178  df-riota 5830  df-sup 6982

This theorem is referenced by:  supsnti  7003  sup3exmid  8912  maxleim  11209  xrmaxleim  11247  supfz  14700