Theorem supcl 4579

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 4578	. . 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 2885	. . 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 10	. 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 4574	. 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 1552	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 223   e. wcel 958  A.wral 1645  E.wrex 1646  E!wreu 1647  {crab 1648  U.cuni 2503   class class class wbr 2619   Or wor 2839  supcsup 4573

This theorem is referenced by:  supub 4580  suplub 4581  supcli 4584  supmax 4589  suprcl 6055  supxrcl 6084  infmxrcl 6086

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-un 2050  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-po 2840  df-so 2850  df-sup 4574

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