Theorem xrlelttr 10040

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 531	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A <_ B /\ B < C ) ) -> A <_ B )
2		simpl1 1026	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A <_ B /\ B < C ) ) -> A e. RR* )
3		simpl2 1027	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A <_ B /\ B < C ) ) -> B e. RR* )
4		xrlenlt 8243	. . . . . 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 624	. . 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 533	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A <_ B /\ B < C ) ) -> B < C )
10		simpl3 1028	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A <_ B /\ B < C ) ) -> C e. RR* )
11		xrltso 10030	. . . . . 6 |- < Or RR*
12		sowlin 4417	. . . . . 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 1273	. . . 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 725	. 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 715   /\ w3a 1004   e. wcel 2202   class class class wbr 4088   Or wor 4392   RR*cxr 8212   < clt 8213   <_ cle 8214

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-po 4393  df-iso 4394  df-xp 4731  df-cnv 4733  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219

This theorem is referenced by:  xrlelttrd  10044  xrre  10054  xrre2  10055  iooss1  10150  iccssioo  10176  iccssico  10179  iocssioo  10197  ioossioo  10199  ico0  10520  bldisj  15124  xblm  15140  blsscls2  15216  metcnpi3  15240