Theorem xrre2 9913

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 9884	. . . . . . 7 |- ( A e. RR* -> -oo <_ A )
2	1	adantr 276	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* ) -> -oo <_ A )
3		mnfxr 8100	. . . . . . 7 |- -oo e. RR*
4		xrlelttr 9898	. . . . . . 7 |- ( ( -oo e. RR* /\ A e. RR* /\ B e. RR* ) -> ( ( -oo <_ A /\ A < B ) -> -oo < B ) )
5	3, 4	mp3an1 1335	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* ) -> ( ( -oo <_ A /\ A < B ) -> -oo < B ) )
6	2, 5	mpand 429	. . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> ( A < B -> -oo < B ) )
7	6	3adant3 1019	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A < B -> -oo < B ) )
8		pnfge 9881	. . . . . . 7 |- ( C e. RR* -> C <_ +oo )
9	8	adantl 277	. . . . . 6 |- ( ( B e. RR* /\ C e. RR* ) -> C <_ +oo )
10		pnfxr 8096	. . . . . . 7 |- +oo e. RR*
11		xrltletr 9899	. . . . . . 7 |- ( ( B e. RR* /\ C e. RR* /\ +oo e. RR* ) -> ( ( B < C /\ C <_ +oo ) -> B < +oo ) )
12	10, 11	mp3an3 1337	. . . . . 6 |- ( ( B e. RR* /\ C e. RR* ) -> ( ( B < C /\ C <_ +oo ) -> B < +oo ) )
13	9, 12	mpan2d 428	. . . . 5 |- ( ( B e. RR* /\ C e. RR* ) -> ( B < C -> B < +oo ) )
14	13	3adant1 1017	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( B < C -> B < +oo ) )
15	7, 14	anim12d 335	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( A < B /\ B < C ) -> ( -oo < B /\ B < +oo ) ) )
16		xrrebnd 9911	. . . 4 |- ( B e. RR* -> ( B e. RR <-> ( -oo < B /\ B < +oo ) ) )
17	16	3ad2ant2 1021	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( B e. RR <-> ( -oo < B /\ B < +oo ) ) )
18	15, 17	sylibrd 169	. 2 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( A < B /\ B < C ) -> B e. RR ) )
19	18	imp 124	1 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < B /\ B < C ) ) -> B e. RR )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   e. wcel 2167   class class class wbr 4034   RRcr 7895   +oocpnf 8075   -oocmnf 8076   RR*cxr 8077   < clt 8078   <_ cle 8079

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-po 4332  df-iso 4333  df-xp 4670  df-cnv 4672  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084

This theorem is referenced by:  elioore  10004  tgioo  14874