Theorem supsnALT 4564

Description: The supremum of a singleton. This version of supsn 4563 is proved directly.

Hypothesis

Ref	Expression
supsn.1	|- R Or A

Assertion

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

Proof of Theorem supsnALT

Step	Hyp	Ref	Expression
1		elsni 2422	. . . . . . 7 |- (y e. {B} -> y = B)
2		breq2 2613	. . . . . . . . 9 |- (y = B -> (BRy <-> BRB))
3	2	negbid 609	. . . . . . . 8 |- (y = B -> (-. BRy <-> -. BRB))
4		supsn.1	. . . . . . . . 9 |- R Or A
5		sonr 2846	. . . . . . . . 9 |- ((R Or A /\ B e. A) -> -. BRB)
6	4, 5	mpan 693	. . . . . . . 8 |- (B e. A -> -. BRB)
7	3, 6	syl5bir 210	. . . . . . 7 |- (y = B -> (B e. A -> -. BRy))
8	1, 7	syl 10	. . . . . 6 |- (y e. {B} -> (B e. A -> -. BRy))
9	8	com12 11	. . . . 5 |- (B e. A -> (y e. {B} -> -. BRy))
10	9	r19.21aiv 1705	. . . 4 |- (B e. A -> A.y e. {B} -. BRy)
11		breq2 2613	. . . . . . . . 9 |- (z = B -> (yRz <-> yRB))
12	11	rcla4ev 1868	. . . . . . . 8 |- ((B e. {B} /\ yRB) -> E.z e. {B}yRz)
13		snidg 2423	. . . . . . . 8 |- (B e. A -> B e. {B})
14	12, 13	sylan 448	. . . . . . 7 |- ((B e. A /\ yRB) -> E.z e. {B}yRz)
15	14	ex 373	. . . . . 6 |- (B e. A -> (yRB -> E.z e. {B}yRz))
16	15	a1d 12	. . . . 5 |- (B e. A -> (y e. A -> (yRB -> E.z e. {B}yRz)))
17	16	r19.21aiv 1705	. . . 4 |- (B e. A -> A.y e. A (yRB -> E.z e. {B}yRz))
18	10, 17	jca 288	. . 3 |- (B e. A -> (A.y e. {B} -. BRy /\ A.y e. A (yRB -> E.z e. {B}yRz)))
19		breq1 2612	. . . . . . . . . . 11 |- (x = B -> (xRy <-> BRy))
20	19	negbid 609	. . . . . . . . . 10 |- (x = B -> (-. xRy <-> -. BRy))
21	20	ralbidv 1655	. . . . . . . . 9 |- (x = B -> (A.y e. {B} -. xRy <-> A.y e. {B} -. BRy))
22		breq2 2613	. . . . . . . . . . 11 |- (x = B -> (yRx <-> yRB))
23	22	imbi1d 611	. . . . . . . . . 10 |- (x = B -> ((yRx -> E.z e. {B}yRz) <-> (yRB -> E.z e. {B}yRz)))
24	23	ralbidv 1655	. . . . . . . . 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 626	. . . . . . . 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 1868	. . . . . . 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 702	. . . . . 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 4550	. . . . . 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 293	. . . . 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 1919	. . . . 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 200	. . . 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 2874	. . . 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 702	. . 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 195	. 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 4548	. 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 1511	1 |- (B e. A -> sup({B}, A, R) = B)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  E*wmo 1374  A.wral 1637  E.wrex 1638  E!wreu 1639  {crab 1640  {csn 2399  U.cuni 2493   class class class wbr 2609   Or wor 2830  supcsup 4547

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-po 2831  df-so 2841  df-sup 4548

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