Theorem poltletr 5066

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 R B /\ B ( R u. _I ) C ) -> A R C ) )

Proof of Theorem poltletr

Step	Hyp	Ref	Expression
1		poleloe 5065	. . . . 5 |- ( C e. X -> ( B ( R u. _I ) C <-> ( B R C \/ B = C ) ) )
2	1	3ad2ant3 1022	. . . 4 |- ( ( A e. X /\ B e. X /\ C e. X ) -> ( B ( R u. _I ) C <-> ( B R C \/ B = C ) ) )
3	2	adantl 277	. . 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 464	. 2 |- ( ( R Po X /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A R B /\ B ( R u. _I ) C ) <-> ( A R B /\ ( B R C \/ B = C ) ) ) )
5		potr 4339	. . . . 5 |- ( ( R Po X /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A R B /\ B R C ) -> A R C ) )
6	5	com12 30	. . . 4 |- ( ( A R B /\ B R C ) -> ( ( R Po X /\ ( A e. X /\ B e. X /\ C e. X ) ) -> A R C ) )
7		breq2 4033	. . . . . 6 |- ( B = C -> ( A R B <-> A R C ) )
8	7	biimpac 298	. . . . 5 |- ( ( A R B /\ B = C ) -> A R C )
9	8	a1d 22	. . . 4 |- ( ( A R B /\ B = C ) -> ( ( R Po X /\ ( A e. X /\ B e. X /\ C e. X ) ) -> A R C ) )
10	6, 9	jaodan 798	. . 3 |- ( ( A R B /\ ( B R C \/ B = C ) ) -> ( ( R Po X /\ ( A e. X /\ B e. X /\ C e. X ) ) -> A R C ) )
11	10	com12 30	. 2 |- ( ( R Po X /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A R B /\ ( B R C \/ B = C ) ) -> A R C ) )
12	4, 11	sylbid 150	1 |- ( ( R Po X /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A R B /\ B ( R u. _I ) C ) -> A R C ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   /\ w3a 980   = wceq 1364   e. wcel 2164   u. cun 3151   class class class wbr 4029   _I cid 4319   Po wpo 4325

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-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-id 4324  df-po 4327  df-xp 4665  df-rel 4666

This theorem is referenced by: (None)