Theorem supmax 5679

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.)

Hypothesis

Ref	Expression
supmax.1	|- R Or A

Assertion

Ref	Expression
supmax	|- ((C e. A /\ C e. B /\ A.y e. B -. CRy) -> sup(B, A, R) = C)

Distinct variable groups:   y,A   y,B   y,C   y,R

Proof of Theorem supmax

Step	Hyp	Ref	Expression
1		supmaxlem 5678	. . . 4 |- ((C e. A /\ C e. B /\ A.y e. B -. CRy) -> E.x e. A (A.z e. B -. xRz /\ A.z e. A (zRx -> E.y e. B zRy)))
2		supmax.1	. . . . 5 |- R Or A
3	2	supcl 5669	. . . 4 |- (E.x e. A (A.z e. B -. xRz /\ A.z e. A (zRx -> E.y e. B zRy)) -> sup(B, A, R) e. A)
4	1, 3	syl 12	. . 3 |- ((C e. A /\ C e. B /\ A.y e. B -. CRy) -> sup(B, A, R) e. A)
5		simp1 876	. . 3 |- ((C e. A /\ C e. B /\ A.y e. B -. CRy) -> C e. A)
6		sotrieq2 3618	. . . 4 |- ((R Or A /\ (sup(B, A, R) e. A /\ C e. A)) -> (sup(B, A, R) = C <-> (-. sup(B, A, R)RC /\ -. CRsup(B, A, R))))
7	2, 6	mpan 759	. . 3 |- ((sup(B, A, R) e. A /\ C e. A) -> (sup(B, A, R) = C <-> (-. sup(B, A, R)RC /\ -. CRsup(B, A, R))))
8	4, 5, 7	syl11anc 524	. 2 |- ((C e. A /\ C e. B /\ A.y e. B -. CRy) -> (sup(B, A, R) = C <-> (-. sup(B, A, R)RC /\ -. CRsup(B, A, R))))
9		simp2 877	. . 3 |- ((C e. A /\ C e. B /\ A.y e. B -. CRy) -> C e. B)
10	2	supub 5670	. . 3 |- (E.x e. A (A.z e. B -. xRz /\ A.z e. A (zRx -> E.y e. B zRy)) -> (C e. B -> -. sup(B, A, R)RC))
11	1, 9, 10	sylc 83	. 2 |- ((C e. A /\ C e. B /\ A.y e. B -. CRy) -> -. sup(B, A, R)RC)
12		3simpb 873	. . 3 |- ((C e. A /\ C e. B /\ A.y e. B -. CRy) -> (C e. A /\ A.y e. B -. CRy))
13	2	supnub 5672	. . 3 |- (E.x e. A (A.z e. B -. xRz /\ A.z e. A (zRx -> E.y e. B zRy)) -> ((C e. A /\ A.y e. B -. CRy) -> -. CRsup(B, A, R)))
14	1, 12, 13	sylc 83	. 2 |- ((C e. A /\ C e. B /\ A.y e. B -. CRy) -> -. CRsup(B, A, R))
15	8, 11, 14	mpbir2and 802	1 |- ((C e. A /\ C e. B /\ A.y e. B -. CRy) -> sup(B, A, R) = C)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106   class class class wbr 3338   Or wor 3590  supcsup 5663

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-po 3591  df-so 3604  df-sup 5664

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