Theorem xrre2 8964

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 8943	. . . . . . 7 |- ( A e. RR* -> -oo <_ A )
2	1	adantr 270	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* ) -> -oo <_ A )
3		mnfxr 7237	. . . . . . 7 |- -oo e. RR*
4		xrlelttr 8952	. . . . . . 7 |- ( ( -oo e. RR* /\ A e. RR* /\ B e. RR* ) -> ( ( -oo <_ A /\ A < B ) -> -oo < B ) )
5	3, 4	mp3an1 1256	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* ) -> ( ( -oo <_ A /\ A < B ) -> -oo < B ) )
6	2, 5	mpand 420	. . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> ( A < B -> -oo < B ) )
7	6	3adant3 959	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A < B -> -oo < B ) )
8		pnfge 8940	. . . . . . 7 |- ( C e. RR* -> C <_ +oo )
9	8	adantl 271	. . . . . 6 |- ( ( B e. RR* /\ C e. RR* ) -> C <_ +oo )
10		pnfxr 7233	. . . . . . 7 |- +oo e. RR*
11		xrltletr 8953	. . . . . . 7 |- ( ( B e. RR* /\ C e. RR* /\ +oo e. RR* ) -> ( ( B < C /\ C <_ +oo ) -> B < +oo ) )
12	10, 11	mp3an3 1258	. . . . . 6 |- ( ( B e. RR* /\ C e. RR* ) -> ( ( B < C /\ C <_ +oo ) -> B < +oo ) )
13	9, 12	mpan2d 419	. . . . 5 |- ( ( B e. RR* /\ C e. RR* ) -> ( B < C -> B < +oo ) )
14	13	3adant1 957	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( B < C -> B < +oo ) )
15	7, 14	anim12d 328	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( A < B /\ B < C ) -> ( -oo < B /\ B < +oo ) ) )
16		xrrebnd 8962	. . . 4 |- ( B e. RR* -> ( B e. RR <-> ( -oo < B /\ B < +oo ) ) )
17	16	3ad2ant2 961	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( B e. RR <-> ( -oo < B /\ B < +oo ) ) )
18	15, 17	sylibrd 167	. 2 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( A < B /\ B < C ) -> B e. RR ) )
19	18	imp 122	1 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < B /\ B < C ) ) -> B e. RR )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   /\ w3a 920   e. wcel 1434   class class class wbr 3793   RRcr 7042   +oocpnf 7212   -oocmnf 7213   RR*cxr 7214   < clt 7215   <_ cle 7216

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-cnex 7129  ax-resscn 7130  ax-pre-ltirr 7150  ax-pre-ltwlin 7151  ax-pre-lttrn 7152

This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-po 4059  df-iso 4060  df-xp 4377  df-cnv 4379  df-pnf 7217  df-mnf 7218  df-xr 7219  df-ltxr 7220  df-le 7221

This theorem is referenced by:  elioore  9011