Theorem supsn 2168

Description: The supremum of a singleton.

Hypothesis

Ref	Expression
supsn.1	|- R Or A

Assertion

Ref	Expression
supsn	|- (B e. A -> sup({B}, A, R) = B)

Proof of Theorem supsn

Step	Hyp	Ref	Expression
1		elsni 1827	. . . . . . 7 |- (y e. {B} -> y = B)
2		breq2 2066	. . . . . . . . 9 |- (y = B -> (BRy <-> BRB))
3	2	negbid 463	. . . . . . . 8 |- (y = B -> (-. BRy <-> -. BRB))
4		supsn.1	. . . . . . . . 9 |- R Or A
5		sonr 2143	. . . . . . . . 9 |- ((R Or A /\ B e. A) -> -. BRB)
6	4, 5	mpan 518	. . . . . . . 8 |- (B e. A -> -. BRB)
7	3, 6	syl5bir 184	. . . . . . 7 |- (y = B -> (B e. A -> -. BRy))
8	1, 7	syl 12	. . . . . 6 |- (y e. {B} -> (B e. A -> -. BRy))
9	8	com12 13	. . . . 5 |- (B e. A -> (y e. {B} -> -. BRy))
10	9	r19.21aiv 1259	. . . 4 |- (B e. A -> A.y e. {B} -. BRy)
11		breq2 2066	. . . . . . . . 9 |- (z = B -> (yRz <-> yRB))
12	11	rcla4ev 1403	. . . . . . . 8 |- ((B e. {B} /\ yRB) -> E.z e. {B}yRz)
13		snidg 1828	. . . . . . . 8 |- (B e. A -> B e. {B})
14	12, 13	sylan 343	. . . . . . 7 |- ((B e. A /\ yRB) -> E.z e. {B}yRz)
15	14	exp 291	. . . . . 6 |- (B e. A -> (yRB -> E.z e. {B}yRz))
16	15	a1d 14	. . . . 5 |- (B e. A -> (y e. A -> (yRB -> E.z e. {B}yRz)))
17	16	r19.21aiv 1259	. . . 4 |- (B e. A -> A.y e. A (yRB -> E.z e. {B}yRz))
18	10, 17	jca 236	. . 3 |- (B e. A -> (A.y e. {B} -. BRy /\ A.y e. A (yRB -> E.z e. {B}yRz)))
19		breq1 2065	. . . . . . . . . . 11 |- (x = B -> (xRy <-> BRy))
20	19	negbid 463	. . . . . . . . . 10 |- (x = B -> (-. xRy <-> -. BRy))
21	20	biraldv 1219	. . . . . . . . 9 |- (x = B -> (A.y e. {B} -. xRy <-> A.y e. {B} -. BRy))
22		breq2 2066	. . . . . . . . . . 11 |- (x = B -> (yRx <-> yRB))
23	22	imbi1d 465	. . . . . . . . . 10 |- (x = B -> ((yRx -> E.z e. {B}yRz) <-> (yRB -> E.z e. {B}yRz)))
24	23	biraldv 1219	. . . . . . . . 9 |- (x = B -> (A.y e. A (yRx -> E.z e. {B}yRz) <-> A.y e. A (yRB -> E.z e. {B}yRz)))
25	21, 24	anbi12d 476	. . . . . . . 8 |- (x = B -> ((A.y e. {B} -. xRy /\ A.y e. A (yRx -> E.z e. {B}yRz)) <-> (A.y e. {B} -. BRy /\ A.y e. A (yRB -> E.z e. {B}yRz))))
26	25	rcla4ev 1403	. . . . . . 7 |- ((B e. A /\ (A.y e. {B} -. BRy /\ A.y e. A (yRB -> E.z e. {B}yRz))) -> E.x e. A (A.y e. {B} -. xRy /\ A.y e. A (yRx -> E.z e. {B}yRz)))
27	18, 26	mpdan 527	. . . . . 6 |- (B e. A -> E.x e. A (A.y e. {B} -. xRy /\ A.y e. A (yRx -> E.z e. {B}yRz)))
28	4	supmo 2156	. . . . . 6 |- E*x(x e. A /\ (A.y e. {B} -. xRy /\ A.y e. A (yRx -> E.z e. {B}yRz)))
29	27, 28	jctir 241	. . . . 5 |- (B e. A -> (E.x e. A (A.y e. {B} -. xRy /\ A.y e. A (yRx -> E.z e. {B}yRz)) /\ E*x(x e. A /\ (A.y e. {B} -. xRy /\ A.y e. A (yRx -> E.z e. {B}yRz)))))
30		reu5 1339	. . . . 5 |- (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)) /\ E*x(x e. A /\ (A.y e. {B} -. xRy /\ A.y e. A (yRx -> E.z e. {B}yRz)))))
31	29, 30	sylibr 175	. . . 4 |- (B e. A -> E!x e. A (A.y e. {B} -. xRy /\ A.y e. A (yRx -> E.z e. {B}yRz)))
32	25	reuuni2 1956	. . . 4 |- ((B 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} -. BRy /\ A.y e. A (yRB -> E.z e. {B}yRz)) <-> U.{x e. A | (A.y e. {B} -. xRy /\ A.y e. A (yRx -> E.z e. {B}yRz))} = B))
33	31, 32	mpdan 527	. . 3 |- (B e. A -> ((A.y e. {B} -. BRy /\ A.y e. A (yRB -> E.z e. {B}yRz)) <-> U.{x e. A | (A.y e. {B} -. xRy /\ A.y e. A (yRx -> E.z e. {B}yRz))} = B))
34	18, 33	mpbid 170	. 2 |- (B e. A -> U.{x e. A | (A.y e. {B} -. xRy /\ A.y e. A (yRx -> E.z e. {B}yRz))} = B)
35		df-sup 2154	. 2 |- sup({B}, A, R) = U.{x e. A | (A.y e. {B} -. xRy /\ A.y e. A (yRx -> E.z e. {B}yRz))}
36	34, 35	syl5eq 1136	1 |- (B e. A -> sup({B}, A, R) = B)

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   /\ wa 196  E*wmo 1008   = wceq 1091   e. wcel 1092  A.wral 1201  E.wrex 1202  E!wreu 1203  {crab 1204  {csn 1808  U.cuni 1919   class class class wbr 2054   Or wor 2059  supcsup 2060

This theorem is referenced by:  sqr0 4730

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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