Theorem xrletr 8724

Description: Transitive law for ordering on extended reals. (Contributed by NM, 9-Feb-2006.)

Assertion

Ref	Expression
xrletr	|- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( A <_ B /\ B <_ C ) -> A <_ C ) )

Proof of Theorem xrletr

Step	Hyp	Ref	Expression
1		xrltso 8717	. . . . . 6 |- < Or RR*
2		sowlin 4057	. . . . . 6 |- ( ( < Or RR* /\ ( C e. RR* /\ A e. RR* /\ B e. RR* ) ) -> ( C < A -> ( C < B \/ B < A ) ) )
3	1, 2	mpan 400	. . . . 5 |- ( ( C e. RR* /\ A e. RR* /\ B e. RR* ) -> ( C < A -> ( C < B \/ B < A ) ) )
4	3	3coml 1111	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( C < A -> ( C < B \/ B < A ) ) )
5		orcom 647	. . . 4 |- ( ( C < B \/ B < A ) <-> ( B < A \/ C < B ) )
6	4, 5	syl6ib 150	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( C < A -> ( B < A \/ C < B ) ) )
7	6	con3d 561	. 2 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( -. ( B < A \/ C < B ) -> -. C < A ) )
8		xrlenlt 7084	. . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> ( A <_ B <-> -. B < A ) )
9	8	3adant3 924	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A <_ B <-> -. B < A ) )
10		xrlenlt 7084	. . . . 5 |- ( ( B e. RR* /\ C e. RR* ) -> ( B <_ C <-> -. C < B ) )
11	10	3adant1 922	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( B <_ C <-> -. C < B ) )
12	9, 11	anbi12d 442	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( A <_ B /\ B <_ C ) <-> ( -. B < A /\ -. C < B ) ) )
13		ioran 669	. . 3 |- ( -. ( B < A \/ C < B ) <-> ( -. B < A /\ -. C < B ) )
14	12, 13	syl6bbr 187	. 2 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( A <_ B /\ B <_ C ) <-> -. ( B < A \/ C < B ) ) )
15		xrlenlt 7084	. . 3 |- ( ( A e. RR* /\ C e. RR* ) -> ( A <_ C <-> -. C < A ) )
16	15	3adant2 923	. 2 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A <_ C <-> -. C < A ) )
17	7, 14, 16	3imtr4d 192	1 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( A <_ B /\ B <_ C ) -> A <_ C ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   <-> wb 98   \/ wo 629   /\ w3a 885   e. wcel 1393   class class class wbr 3764   Or wor 4032   RR*cxr 7059   < clt 7060   <_ cle 7061

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-cnex 6975  ax-resscn 6976  ax-pre-ltirr 6996  ax-pre-ltwlin 6997  ax-pre-lttrn 6998

This theorem depends on definitions:  df-bi 110  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-po 4033  df-iso 4034  df-xp 4351  df-cnv 4353  df-pnf 7062  df-mnf 7063  df-xr 7064  df-ltxr 7065  df-le 7066

This theorem is referenced by:  xrletrd  8728  icc0r  8795  iccss  8810  icossico  8812  iccss2  8813  iccssico  8814