Theorem supsnALT 4572

Description: The supremum of a singleton. This version of supsn 4571 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 2428	. . . . . . 7 |- (y e. {B} -> y = B)
2		breq2 2618	. . . . . . . . 9 |- (y = B -> (BRy <-> BRB))
3	2	negbid 610	. . . . . . . 8 |- (y = B -> (-. BRy <-> -. BRB))
4		supsn.1	. . . . . . . . 9 |- R Or A
5		sonr 2850	. . . . . . . . 9 |- ((R Or A /\ B e. A) -> -. BRB)
6	4, 5	mpan 694	. . . . . . . 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 1710	. . . 4 |- (B e. A -> A.y e. {B} -. BRy)
11		breq2 2618	. . . . . . . . 9 |- (z = B -> (yRz <-> yRB))
12	11	rcla4ev 1873	. . . . . . . 8 |- ((B e. {B} /\ yRB) -> E.z e. {B}yRz)
13		snidg 2429	. . . . . . . 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 1710	. . . 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 2617	. . . . . . . . . . 11 |- (x = B -> (xRy <-> BRy))
20	19	negbid 610	. . . . . . . . . 10 |- (x = B -> (-. xRy <-> -. BRy))
21	20	ralbidv 1660	. . . . . . . . 9 |- (x = B -> (A.y e. {B} -. xRy <-> A.y e. {B} -. BRy))
22		breq2 2618	. . . . . . . . . . 11 |- (x = B -> (yRx <-> yRB))
23	22	imbi1d 612	. . . . . . . . . 10 |- (x = B -> ((yRx -> E.z e. {B}yRz) <-> (yRB -> E.z e. {B}yRz)))
24	23	ralbidv 1660	. . . . . . . . 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 627	. . . . . . . 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 1873	. . . . . . 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 703	. . . . . 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 4556	. . . . . 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 1925	. . . . 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 2879	. . . 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 703	. . 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 4554	. 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 1516	1 |- (B e. A -> sup({B}, A, R) = B)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   = wceq 954   e. wcel 956  E*wmo 1379  A.wral 1642  E.wrex 1643  E!wreu 1644  {crab 1645  {csn 2405  U.cuni 2498   class class class wbr 2614   Or wor 2834  supcsup 4553

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-reu 1648  df-rab 1649  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-po 2835  df-so 2845  df-sup 4554

Copyright terms: Public domain