Theorem supmax 4598

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		supmax.1	. . . . 5 |- R Or A
2	1	supub 4589	. . . 4 |- (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))
3		supmaxlem 4597	. . . 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)))
4		3simp2 791	. . . 4 |- ((C e. A /\ C e. B /\ A.y e. B -. CRy) -> C e. B)
5	2, 3, 4	sylc 68	. . 3 |- ((C e. A /\ C e. B /\ A.y e. B -. CRy) -> -. sup(B, A, R)RC)
6	1	supnub 4591	. . . 4 |- (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)))
7		3simpb 788	. . . 4 |- ((C e. A /\ C e. B /\ A.y e. B -. CRy) -> (C e. A /\ A.y e. B -. CRy))
8	6, 3, 7	sylc 68	. . 3 |- ((C e. A /\ C e. B /\ A.y e. B -. CRy) -> -. CRsup(B, A, R))
9	5, 8	jca 288	. 2 |- ((C e. A /\ C e. B /\ A.y e. B -. CRy) -> (-. sup(B, A, R)RC /\ -. CRsup(B, A, R)))
10		sotrieq2 2868	. . . 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))))
11	1, 10	mpan 697	. . 3 |- ((sup(B, A, R) e. A /\ C e. A) -> (sup(B, A, R) = C <-> (-. sup(B, A, R)RC /\ -. CRsup(B, A, R))))
12	1	supcl 4588	. . . 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)
13	3, 12	syl 10	. . 3 |- ((C e. A /\ C e. B /\ A.y e. B -. CRy) -> sup(B, A, R) e. A)
14		3simp1 790	. . 3 |- ((C e. A /\ C e. B /\ A.y e. B -. CRy) -> C e. A)
15	11, 13, 14	sylanc 473	. 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))))
16	9, 15	mpbird 196	1 |- ((C e. A /\ C e. B /\ A.y e. B -. CRy) -> sup(B, A, R) = C)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 777   = wceq 958   e. wcel 960  A.wral 1648  E.wrex 1649   class class class wbr 2624   Or wor 2845  supcsup 4582

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-po 2846  df-so 2856  df-sup 4583

Copyright terms: Public domain