Theorem pocl 2132

Description: Properties of partial order relation in class notation.

Assertion

Ref	Expression
pocl	|- (R Po A -> ((B e. A /\ C e. A /\ D e. A) -> (-. BRB /\ ((BRC /\ CRD) -> BRD))))

Proof of Theorem pocl

Step	Hyp	Ref	Expression
1		id 9	. . . . . . 7 |- (x = B -> x = B)
2	1, 1	breq12d 2073	. . . . . 6 |- (x = B -> (xRx <-> BRB))
3	2	negbid 463	. . . . 5 |- (x = B -> (-. xRx <-> -. BRB))
4		breq1 2065	. . . . . . 7 |- (x = B -> (xRy <-> BRy))
5	4	anbi1d 469	. . . . . 6 |- (x = B -> ((xRy /\ yRz) <-> (BRy /\ yRz)))
6		breq1 2065	. . . . . 6 |- (x = B -> (xRz <-> BRz))
7	5, 6	imbi12d 474	. . . . 5 |- (x = B -> (((xRy /\ yRz) -> xRz) <-> ((BRy /\ yRz) -> BRz)))
8	3, 7	anbi12d 476	. . . 4 |- (x = B -> ((-. xRx /\ ((xRy /\ yRz) -> xRz)) <-> (-. BRB /\ ((BRy /\ yRz) -> BRz))))
9	8	imbi2d 464	. . 3 |- (x = B -> ((R Po A -> (-. xRx /\ ((xRy /\ yRz) -> xRz))) <-> (R Po A -> (-. BRB /\ ((BRy /\ yRz) -> BRz)))))
10		breq2 2066	. . . . . . 7 |- (y = C -> (BRy <-> BRC))
11		breq1 2065	. . . . . . 7 |- (y = C -> (yRz <-> CRz))
12	10, 11	anbi12d 476	. . . . . 6 |- (y = C -> ((BRy /\ yRz) <-> (BRC /\ CRz)))
13	12	imbi1d 465	. . . . 5 |- (y = C -> (((BRy /\ yRz) -> BRz) <-> ((BRC /\ CRz) -> BRz)))
14	13	anbi2d 468	. . . 4 |- (y = C -> ((-. BRB /\ ((BRy /\ yRz) -> BRz)) <-> (-. BRB /\ ((BRC /\ CRz) -> BRz))))
15	14	imbi2d 464	. . 3 |- (y = C -> ((R Po A -> (-. BRB /\ ((BRy /\ yRz) -> BRz))) <-> (R Po A -> (-. BRB /\ ((BRC /\ CRz) -> BRz)))))
16		breq2 2066	. . . . . . 7 |- (z = D -> (CRz <-> CRD))
17	16	anbi2d 468	. . . . . 6 |- (z = D -> ((BRC /\ CRz) <-> (BRC /\ CRD)))
18		breq2 2066	. . . . . 6 |- (z = D -> (BRz <-> BRD))
19	17, 18	imbi12d 474	. . . . 5 |- (z = D -> (((BRC /\ CRz) -> BRz) <-> ((BRC /\ CRD) -> BRD)))
20	19	anbi2d 468	. . . 4 |- (z = D -> ((-. BRB /\ ((BRC /\ CRz) -> BRz)) <-> (-. BRB /\ ((BRC /\ CRD) -> BRD))))
21	20	imbi2d 464	. . 3 |- (z = D -> ((R Po A -> (-. BRB /\ ((BRC /\ CRz) -> BRz))) <-> (R Po A -> (-. BRB /\ ((BRC /\ CRD) -> BRD)))))
22		df-po 2128	. . . . . . . 8 |- (R Po A <-> A.x e. A A.y e. A A.z e. A (-. xRx /\ ((xRy /\ yRz) -> xRz)))
23		r3al 1240	. . . . . . . 8 |- (A.x e. A A.y e. A A.z e. A (-. xRx /\ ((xRy /\ yRz) -> xRz)) <-> A.xA.yA.z((x e. A /\ y e. A /\ z e. A) -> (-. xRx /\ ((xRy /\ yRz) -> xRz))))
24	22, 23	bitr 151	. . . . . . 7 |- (R Po A <-> A.xA.yA.z((x e. A /\ y e. A /\ z e. A) -> (-. xRx /\ ((xRy /\ yRz) -> xRz))))
25	24	biimp 133	. . . . . 6 |- (R Po A -> A.xA.yA.z((x e. A /\ y e. A /\ z e. A) -> (-. xRx /\ ((xRy /\ yRz) -> xRz))))
26	25	19.21bbi 743	. . . . 5 |- (R Po A -> A.z((x e. A /\ y e. A /\ z e. A) -> (-. xRx /\ ((xRy /\ yRz) -> xRz))))
27	26	19.21bi 742	. . . 4 |- (R Po A -> ((x e. A /\ y e. A /\ z e. A) -> (-. xRx /\ ((xRy /\ yRz) -> xRz))))
28	27	com12 13	. . 3 |- ((x e. A /\ y e. A /\ z e. A) -> (R Po A -> (-. xRx /\ ((xRy /\ yRz) -> xRz))))
29	9, 15, 21, 28	vtocl3ga 1389	. 2 |- ((B e. A /\ C e. A /\ D e. A) -> (R Po A -> (-. BRB /\ ((BRC /\ CRD) -> BRD))))
30	29	com12 13	1 |- (R Po A -> ((B e. A /\ C e. A /\ D e. A) -> (-. BRB /\ ((BRC /\ CRD) -> BRD))))

Syntax hints:  -. wn 1   -> wi 2   /\ wa 196   /\ w3a 581  A.wal 672   = wceq 1091   e. wcel 1092  A.wral 1201   class class class wbr 2054   Po wpo 2058

This theorem is referenced by:  poirr 2133  potr 2134

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-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-v 1349  df-un 1490  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-po 2128

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