Theorem pocl 4321

Description: Properties of partial order relation in class notation. (Contributed by NM, 27-Mar-1997.)

Assertion

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

Proof of Theorem pocl

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 19	. . . . . . 7 |- ( x = B -> x = B )
2	1, 1	breq12d 4031	. . . . . 6 |- ( x = B -> ( x R x <-> B R B ) )
3	2	notbid 668	. . . . 5 |- ( x = B -> ( -. x R x <-> -. B R B ) )
4		breq1 4021	. . . . . . 7 |- ( x = B -> ( x R y <-> B R y ) )
5	4	anbi1d 465	. . . . . 6 |- ( x = B -> ( ( x R y /\ y R z ) <-> ( B R y /\ y R z ) ) )
6		breq1 4021	. . . . . 6 |- ( x = B -> ( x R z <-> B R z ) )
7	5, 6	imbi12d 234	. . . . 5 |- ( x = B -> ( ( ( x R y /\ y R z ) -> x R z ) <-> ( ( B R y /\ y R z ) -> B R z ) ) )
8	3, 7	anbi12d 473	. . . 4 |- ( x = B -> ( ( -. x R x /\ ( ( x R y /\ y R z ) -> x R z ) ) <-> ( -. B R B /\ ( ( B R y /\ y R z ) -> B R z ) ) ) )
9	8	imbi2d 230	. . 3 |- ( x = B -> ( ( R Po A -> ( -. x R x /\ ( ( x R y /\ y R z ) -> x R z ) ) ) <-> ( R Po A -> ( -. B R B /\ ( ( B R y /\ y R z ) -> B R z ) ) ) ) )
10		breq2 4022	. . . . . . 7 |- ( y = C -> ( B R y <-> B R C ) )
11		breq1 4021	. . . . . . 7 |- ( y = C -> ( y R z <-> C R z ) )
12	10, 11	anbi12d 473	. . . . . 6 |- ( y = C -> ( ( B R y /\ y R z ) <-> ( B R C /\ C R z ) ) )
13	12	imbi1d 231	. . . . 5 |- ( y = C -> ( ( ( B R y /\ y R z ) -> B R z ) <-> ( ( B R C /\ C R z ) -> B R z ) ) )
14	13	anbi2d 464	. . . 4 |- ( y = C -> ( ( -. B R B /\ ( ( B R y /\ y R z ) -> B R z ) ) <-> ( -. B R B /\ ( ( B R C /\ C R z ) -> B R z ) ) ) )
15	14	imbi2d 230	. . 3 |- ( y = C -> ( ( R Po A -> ( -. B R B /\ ( ( B R y /\ y R z ) -> B R z ) ) ) <-> ( R Po A -> ( -. B R B /\ ( ( B R C /\ C R z ) -> B R z ) ) ) ) )
16		breq2 4022	. . . . . . 7 |- ( z = D -> ( C R z <-> C R D ) )
17	16	anbi2d 464	. . . . . 6 |- ( z = D -> ( ( B R C /\ C R z ) <-> ( B R C /\ C R D ) ) )
18		breq2 4022	. . . . . 6 |- ( z = D -> ( B R z <-> B R D ) )
19	17, 18	imbi12d 234	. . . . 5 |- ( z = D -> ( ( ( B R C /\ C R z ) -> B R z ) <-> ( ( B R C /\ C R D ) -> B R D ) ) )
20	19	anbi2d 464	. . . 4 |- ( z = D -> ( ( -. B R B /\ ( ( B R C /\ C R z ) -> B R z ) ) <-> ( -. B R B /\ ( ( B R C /\ C R D ) -> B R D ) ) ) )
21	20	imbi2d 230	. . 3 |- ( z = D -> ( ( R Po A -> ( -. B R B /\ ( ( B R C /\ C R z ) -> B R z ) ) ) <-> ( R Po A -> ( -. B R B /\ ( ( B R C /\ C R D ) -> B R D ) ) ) ) )
22		df-po 4314	. . . . . . . 8 |- ( R Po A <-> A. x e. A A. y e. A A. z e. A ( -. x R x /\ ( ( x R y /\ y R z ) -> x R z ) ) )
23		r3al 2534	. . . . . . . 8 |- ( A. x e. A A. y e. A A. z e. A ( -. x R x /\ ( ( x R y /\ y R z ) -> x R z ) ) <-> A. x A. y A. z ( ( x e. A /\ y e. A /\ z e. A ) -> ( -. x R x /\ ( ( x R y /\ y R z ) -> x R z ) ) ) )
24	22, 23	bitri 184	. . . . . . 7 |- ( R Po A <-> A. x A. y A. z ( ( x e. A /\ y e. A /\ z e. A ) -> ( -. x R x /\ ( ( x R y /\ y R z ) -> x R z ) ) ) )
25	24	biimpi 120	. . . . . 6 |- ( R Po A -> A. x A. y A. z ( ( x e. A /\ y e. A /\ z e. A ) -> ( -. x R x /\ ( ( x R y /\ y R z ) -> x R z ) ) ) )
26	25	19.21bbi 1570	. . . . 5 |- ( R Po A -> A. z ( ( x e. A /\ y e. A /\ z e. A ) -> ( -. x R x /\ ( ( x R y /\ y R z ) -> x R z ) ) ) )
27	26	19.21bi 1569	. . . 4 |- ( R Po A -> ( ( x e. A /\ y e. A /\ z e. A ) -> ( -. x R x /\ ( ( x R y /\ y R z ) -> x R z ) ) ) )
28	27	com12 30	. . 3 |- ( ( x e. A /\ y e. A /\ z e. A ) -> ( R Po A -> ( -. x R x /\ ( ( x R y /\ y R z ) -> x R z ) ) ) )
29	9, 15, 21, 28	vtocl3ga 2822	. 2 |- ( ( B e. A /\ C e. A /\ D e. A ) -> ( R Po A -> ( -. B R B /\ ( ( B R C /\ C R D ) -> B R D ) ) ) )
30	29	com12 30	1 |- ( R Po A -> ( ( B e. A /\ C e. A /\ D e. A ) -> ( -. B R B /\ ( ( B R C /\ C R D ) -> B R D ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   /\ w3a 980   A.wal 1362   = wceq 1364   e. wcel 2160   A.wral 2468   class class class wbr 4018   Po wpo 4312

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-po 4314

This theorem is referenced by:  poirr  4325  potr  4326