Theorem poltletr 5083

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 5082	. . . . 5 |- ( C e. X -> ( B ( R u. _I ) C <-> ( B R C \/ B = C ) ) )
2	1	3ad2ant3 1023	. . . 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 4355	. . . . 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 4048	. . . . . 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 799	. . 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 710   /\ w3a 981   = wceq 1373   e. wcel 2176   u. cun 3164   class class class wbr 4044   _I cid 4335   Po wpo 4341

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-br 4045  df-opab 4106  df-id 4340  df-po 4343  df-xp 4681  df-rel 4682

This theorem is referenced by: (None)