Theorem xrre2 9604

Description: An extended real between two others is real. (Contributed by NM, 6-Feb-2007.)

Assertion

Ref	Expression
xrre2	|- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < B /\ B < C ) ) -> B e. RR )

Proof of Theorem xrre2

Step	Hyp	Ref	Expression
1		mnfle 9578	. . . . . . 7 |- ( A e. RR* -> -oo <_ A )
2	1	adantr 274	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* ) -> -oo <_ A )
3		mnfxr 7822	. . . . . . 7 |- -oo e. RR*
4		xrlelttr 9589	. . . . . . 7 |- ( ( -oo e. RR* /\ A e. RR* /\ B e. RR* ) -> ( ( -oo <_ A /\ A < B ) -> -oo < B ) )
5	3, 4	mp3an1 1302	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* ) -> ( ( -oo <_ A /\ A < B ) -> -oo < B ) )
6	2, 5	mpand 425	. . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> ( A < B -> -oo < B ) )
7	6	3adant3 1001	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A < B -> -oo < B ) )
8		pnfge 9575	. . . . . . 7 |- ( C e. RR* -> C <_ +oo )
9	8	adantl 275	. . . . . 6 |- ( ( B e. RR* /\ C e. RR* ) -> C <_ +oo )
10		pnfxr 7818	. . . . . . 7 |- +oo e. RR*
11		xrltletr 9590	. . . . . . 7 |- ( ( B e. RR* /\ C e. RR* /\ +oo e. RR* ) -> ( ( B < C /\ C <_ +oo ) -> B < +oo ) )
12	10, 11	mp3an3 1304	. . . . . 6 |- ( ( B e. RR* /\ C e. RR* ) -> ( ( B < C /\ C <_ +oo ) -> B < +oo ) )
13	9, 12	mpan2d 424	. . . . 5 |- ( ( B e. RR* /\ C e. RR* ) -> ( B < C -> B < +oo ) )
14	13	3adant1 999	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( B < C -> B < +oo ) )
15	7, 14	anim12d 333	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( A < B /\ B < C ) -> ( -oo < B /\ B < +oo ) ) )
16		xrrebnd 9602	. . . 4 |- ( B e. RR* -> ( B e. RR <-> ( -oo < B /\ B < +oo ) ) )
17	16	3ad2ant2 1003	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( B e. RR <-> ( -oo < B /\ B < +oo ) ) )
18	15, 17	sylibrd 168	. 2 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( A < B /\ B < C ) -> B e. RR ) )
19	18	imp 123	1 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < B /\ B < C ) ) -> B e. RR )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 962   e. wcel 1480   class class class wbr 3929   RRcr 7619   +oocpnf 7797   -oocmnf 7798   RR*cxr 7799   < clt 7800   <_ cle 7801

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734

This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-po 4218  df-iso 4219  df-xp 4545  df-cnv 4547  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806

This theorem is referenced by:  elioore  9695  tgioo  12715