Theorem supcl 2159

Description: A supremum belongs to its base class (closure law).

Hypothesis

Ref	Expression
supmo.1	|- R Or A

Assertion

Ref	Expression
supcl	|- (E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)) -> sup(B, A, R) e. A)

Distinct variable group(s):   x,y,z,A   x,R,y,z   x,B,y,z

Proof of Theorem supcl

Step	Hyp	Ref	Expression
1		supmo.1	. . . 4 |- R Or A
2	1	supeu 2158	. . 3 |- (E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)) -> E!x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)))
3		reucl 1957	. . 3 |- (E!x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)) -> U.{x e. A | (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz))} e. A)
4	2, 3	syl 12	. 2 |- (E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)) -> U.{x e. A | (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz))} e. A)
5		df-sup 2154	. . 3 |- sup(B, A, R) = U.{x e. A | (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz))}
6	5	eleq1i 1152	. 2 |- (sup(B, A, R) e. A <-> U.{x e. A | (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz))} e. A)
7	4, 6	sylibr 175	1 |- (E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)) -> sup(B, A, R) e. A)

Syntax hints:  -. wn 1   -> wi 2   /\ wa 196   e. wcel 1092  A.wral 1201  E.wrex 1202  E!wreu 1203  {crab 1204  U.cuni 1919   class class class wbr 2054   Or wor 2059  supcsup 2060

This theorem is referenced by:  supub 2160  suplub 2161  supcli 2164  suprcl 4512

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-un 1490  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-po 2128  df-so 2138  df-sup 2154

metamath.org