Theorem supcl 5669

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 groups:   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 5668	. . 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 3213	. . 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 5664	. 2 |- sup(B, A, R) = U.{x e. A | (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz))}
6	4, 5	syl5eqel 1975	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 2   -> wi 3   /\ wa 240   e. wcel 1300  A.wral 2105  E.wrex 2106  E!wreu 2107  {crab 2108  U.cuni 3177   class class class wbr 3338   Or wor 3590  supcsup 5663

This theorem is referenced by:  supub 5670  suplub 5671  supcli 5674  supmax 5679  suprcl 7264  supxrcl 7293  infmxrcl 7295  supclt 15762

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-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

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-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-un 2600  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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