Theorem poltletr 5070

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 5069	. . . . 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 4343	. . . . 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 4037	. . . . . 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 2167   u. cun 3155   class class class wbr 4033   _I cid 4323   Po wpo 4329

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-id 4328  df-po 4331  df-xp 4669  df-rel 4670

This theorem is referenced by: (None)