Theorem supub 2160

Description: A supremum is an upper bound.

Hypothesis

Ref	Expression
supmo.1	|- R Or A

Assertion

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

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

Proof of Theorem supub

Step	Hyp	Ref	Expression
1		breq2 2066	. . . . 5 |- (w = C -> (sup(B, A, R)Rw <-> sup(B, A, R)RC))
2	1	negbid 463	. . . 4 |- (w = C -> (-. sup(B, A, R)Rw <-> -. sup(B, A, R)RC))
3	2	imbi2d 464	. . 3 |- (w = C -> ((E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)) -> -. sup(B, A, R)Rw) <-> (E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)) -> -. sup(B, A, R)RC)))
4		df-sup 2154	. . . . . . . 8 |- sup(B, A, R) = U.{x e. A | (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz))}
5	4	cleqcomi 1105	. . . . . . 7 |- U.{x e. A | (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz))} = sup(B, A, R)
6		breq1 2065	. . . . . . . . . . . 12 |- (x = sup(B, A, R) -> (xRy <-> sup(B, A, R)Ry))
7	6	negbid 463	. . . . . . . . . . 11 |- (x = sup(B, A, R) -> (-. xRy <-> -. sup(B, A, R)Ry))
8	7	biraldv 1219	. . . . . . . . . 10 |- (x = sup(B, A, R) -> (A.y e. B -. xRy <-> A.y e. B -. sup(B, A, R)Ry))
9		breq2 2066	. . . . . . . . . . . 12 |- (x = sup(B, A, R) -> (yRx <-> yRsup(B, A, R)))
10	9	imbi1d 465	. . . . . . . . . . 11 |- (x = sup(B, A, R) -> ((yRx -> E.z e. B yRz) <-> (yRsup(B, A, R) -> E.z e. B yRz)))
11	10	biraldv 1219	. . . . . . . . . 10 |- (x = sup(B, A, R) -> (A.y e. A (yRx -> E.z e. B yRz) <-> A.y e. A (yRsup(B, A, R) -> E.z e. B yRz)))
12	8, 11	anbi12d 476	. . . . . . . . 9 |- (x = sup(B, A, R) -> ((A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)) <-> (A.y e. B -. sup(B, A, R)Ry /\ A.y e. A (yRsup(B, A, R) -> E.z e. B yRz))))
13	12	reuuni2 1956	. . . . . . . 8 |- ((sup(B, A, R) e. A /\ E!x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz))) -> ((A.y e. B -. sup(B, A, R)Ry /\ A.y e. A (yRsup(B, A, R) -> E.z e. B yRz)) <-> U.{x e. A | (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz))} = sup(B, A, R)))
14		supmo.1	. . . . . . . . 9 |- R Or A
15	14	supcl 2159	. . . . . . . 8 |- (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)
16	14	supeu 2158	. . . . . . . 8 |- (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)))
17	13, 15, 16	sylanc 361	. . . . . . 7 |- (E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)) -> ((A.y e. B -. sup(B, A, R)Ry /\ A.y e. A (yRsup(B, A, R) -> E.z e. B yRz)) <-> U.{x e. A | (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz))} = sup(B, A, R)))
18	5, 17	mpbiri 169	. . . . . 6 |- (E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)) -> (A.y e. B -. sup(B, A, R)Ry /\ A.y e. A (yRsup(B, A, R) -> E.z e. B yRz)))
19	18	pm3.26d 258	. . . . 5 |- (E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)) -> A.y e. B -. sup(B, A, R)Ry)
20		breq2 2066	. . . . . . 7 |- (y = w -> (sup(B, A, R)Ry <-> sup(B, A, R)Rw))
21	20	negbid 463	. . . . . 6 |- (y = w -> (-. sup(B, A, R)Ry <-> -. sup(B, A, R)Rw))
22	21	rcla4v 1402	. . . . 5 |- (A.y e. B -. sup(B, A, R)Ry -> (w e. B -> -. sup(B, A, R)Rw))
23	19, 22	syl 12	. . . 4 |- (E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)) -> (w e. B -> -. sup(B, A, R)Rw))
24	23	com12 13	. . 3 |- (w e. B -> (E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)) -> -. sup(B, A, R)Rw))
25	3, 24	vtoclga 1387	. 2 |- (C e. B -> (E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)) -> -. sup(B, A, R)RC))
26	25	com12 13	1 |- (E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)) -> (C e. B -> -. sup(B, A, R)RC))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   /\ wa 196   = weq 797   = wceq 1091   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:  supubi 2165  suprub 4513

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-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

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-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-po 2128  df-so 2138  df-sup 2154

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