Theorem poltletr 4668

Description: Transitive law for general strict orders. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Assertion

Ref	Expression
poltletr	|- (( R Po X /\ (A e. X /\ B e. X /\ C e. X)) -> ((A RB /\ B( R u. _I ) C) -> A R C))

Proof of Theorem poltletr

Step	Hyp	Ref	Expression
1		poleloe 4667	. . . . 5 |- ( C e. X -> (B( R u. _I ) C <-> (B R C \/ B = C)))
2	1	3ad2ant3 926	. . . 4 |- ((A e. X /\ B e. X /\ C e. X) -> (B( R u. _I ) C <-> (B R C \/ B = C)))
3	2	adantl 262	. . 3 |- (( R Po X /\ (A e. X /\ B e. X /\ C e. X)) -> (B( R u. _I ) C <-> (B R C \/ B = C)))
4	3	anbi2d 437	. 2 |- (( R Po X /\ (A e. X /\ B e. X /\ C e. X)) -> ((A RB /\ B( R u. _I ) C) <-> (A RB /\ (B R C \/ B = C))))
5		potr 4036	. . . . 5 |- (( R Po X /\ (A e. X /\ B e. X /\ C e. X)) -> ((A RB /\ B R C) -> A R C))
6	5	com12 27	. . . 4 |- ((A RB /\ B R C) -> (( R Po X /\ (A e. X /\ B e. X /\ C e. X)) -> A R C))
7		breq2 3759	. . . . . 6 |- (B = C -> (A RB <-> A R C))
8	7	biimpac 282	. . . . 5 |- ((A RB /\ B = C) -> A R C)
9	8	a1d 22	. . . 4 |- ((A RB /\ B = C) -> (( R Po X /\ (A e. X /\ B e. X /\ C e. X)) -> A R C))
10	6, 9	jaodan 709	. . 3 |- ((A RB /\ (B R C \/ B = C)) -> (( R Po X /\ (A e. X /\ B e. X /\ C e. X)) -> A R C))
11	10	com12 27	. 2 |- (( R Po X /\ (A e. X /\ B e. X /\ C e. X)) -> ((A RB /\ (B R C \/ B = C)) -> A R C))
12	4, 11	sylbid 139	1 |- (( R Po X /\ (A e. X /\ B e. X /\ C e. X)) -> ((A RB /\ B( R u. _I ) C) -> A R C))

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   \/ wo 628   /\ w3a 884   = wceq 1242   e. wcel 1390   u. cun 2909   class class class wbr 3755   _I cid 4016   Po wpo 4022

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-po 4024  df-xp 4294  df-rel 4295

This theorem is referenced by: (None)