Theorem xrlelttr 9808

Description: Transitive law for ordering on extended reals. (Contributed by NM, 19-Jan-2006.)

Assertion

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

Proof of Theorem xrlelttr

Step	Hyp	Ref	Expression
1		simprl 529	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A <_ B /\ B < C ) ) -> A <_ B )
2		simpl1 1000	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A <_ B /\ B < C ) ) -> A e. RR* )
3		simpl2 1001	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A <_ B /\ B < C ) ) -> B e. RR* )
4		xrlenlt 8024	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* ) -> ( A <_ B <-> -. B < A ) )
5	2, 3, 4	syl2anc 411	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A <_ B /\ B < C ) ) -> ( A <_ B <-> -. B < A ) )
6	1, 5	mpbid 147	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A <_ B /\ B < C ) ) -> -. B < A )
7	6	pm2.21d 619	. . 3 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A <_ B /\ B < C ) ) -> ( B < A -> A < C ) )
8		idd 21	. . 3 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A <_ B /\ B < C ) ) -> ( A < C -> A < C ) )
9		simprr 531	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A <_ B /\ B < C ) ) -> B < C )
10		simpl3 1002	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A <_ B /\ B < C ) ) -> C e. RR* )
11		xrltso 9798	. . . . . 6 |- < Or RR*
12		sowlin 4322	. . . . . 6 |- ( ( < Or RR* /\ ( B e. RR* /\ C e. RR* /\ A e. RR* ) ) -> ( B < C -> ( B < A \/ A < C ) ) )
13	11, 12	mpan 424	. . . . 5 |- ( ( B e. RR* /\ C e. RR* /\ A e. RR* ) -> ( B < C -> ( B < A \/ A < C ) ) )
14	3, 10, 2, 13	syl3anc 1238	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A <_ B /\ B < C ) ) -> ( B < C -> ( B < A \/ A < C ) ) )
15	9, 14	mpd 13	. . 3 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A <_ B /\ B < C ) ) -> ( B < A \/ A < C ) )
16	7, 8, 15	mpjaod 718	. 2 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A <_ B /\ B < C ) ) -> A < C )
17	16	ex 115	1 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( A <_ B /\ B < C ) -> A < C ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 708   /\ w3a 978   e. wcel 2148   class class class wbr 4005   Or wor 4297   RR*cxr 7993   < clt 7994   <_ cle 7995

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-po 4298  df-iso 4299  df-xp 4634  df-cnv 4636  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000

This theorem is referenced by:  xrlelttrd  9812  xrre  9822  xrre2  9823  iooss1  9918  iccssioo  9944  iccssico  9947  iocssioo  9965  ioossioo  9967  ico0  10264  bldisj  13940  xblm  13956  blsscls2  14032  metcnpi3  14056