Theorem poltletr 4979

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 4978	. . . . 5 |- ( C e. X -> ( B ( R u. _I ) C <-> ( B R C \/ B = C ) ) )
2	1	3ad2ant3 1005	. . . 4 |- ( ( A e. X /\ B e. X /\ C e. X ) -> ( B ( R u. _I ) C <-> ( B R C \/ B = C ) ) )
3	2	adantl 275	. . 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 460	. 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 4263	. . . . 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 3965	. . . . . 6 |- ( B = C -> ( A R B <-> A R C ) )
8	7	biimpac 296	. . . . 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 787	. . 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 149	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 103   <-> wb 104   \/ wo 698   /\ w3a 963   = wceq 1332   e. wcel 2125   u. cun 3096   class class class wbr 3961   _I cid 4243   Po wpo 4249

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-pow 4130  ax-pr 4164

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ral 2437  df-rex 2438  df-v 2711  df-un 3102  df-in 3104  df-ss 3111  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-br 3962  df-opab 4022  df-id 4248  df-po 4251  df-xp 4585  df-rel 4586

This theorem is referenced by: (None)