Theorem poltletr 5137

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 5136	. . . . 5 |- ( C e. X -> ( B ( R u. _I ) C <-> ( B R C \/ B = C ) ) )
2	1	3ad2ant3 1046	. . . 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 4405	. . . . 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 4092	. . . . . 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 804	. . 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 715   /\ w3a 1004   = wceq 1397   e. wcel 2202   u. cun 3198   class class class wbr 4088   _I cid 4385   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-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-id 4390  df-po 4393  df-xp 4731  df-rel 4732

This theorem is referenced by: (None)