Theorem pocl 4400

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 4101	. . . . . 6 |- ( x = B -> ( x R x <-> B R B ) )
3	2	notbid 673	. . . . 5 |- ( x = B -> ( -. x R x <-> -. B R B ) )
4		breq1 4091	. . . . . . 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 4091	. . . . . 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 4092	. . . . . . 7 |- ( y = C -> ( B R y <-> B R C ) )
11		breq1 4091	. . . . . . 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 4092	. . . . . . 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 4092	. . . . . 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 4393	. . . . . . . 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 2576	. . . . . . . 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 1607	. . . . 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 1606	. . . 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 2874	. 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 1004   A.wal 1395   = wceq 1397   e. wcel 2202   A.wral 2510   class class class wbr 4088   Po wpo 4391

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-po 4393

This theorem is referenced by:  poirr  4404  potr  4405