Theorem xrre2 9778

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 9749	. . . . . . 7 |- ( A e. RR* -> -oo <_ A )
2	1	adantr 274	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* ) -> -oo <_ A )
3		mnfxr 7976	. . . . . . 7 |- -oo e. RR*
4		xrlelttr 9763	. . . . . . 7 |- ( ( -oo e. RR* /\ A e. RR* /\ B e. RR* ) -> ( ( -oo <_ A /\ A < B ) -> -oo < B ) )
5	3, 4	mp3an1 1319	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* ) -> ( ( -oo <_ A /\ A < B ) -> -oo < B ) )
6	2, 5	mpand 427	. . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> ( A < B -> -oo < B ) )
7	6	3adant3 1012	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A < B -> -oo < B ) )
8		pnfge 9746	. . . . . . 7 |- ( C e. RR* -> C <_ +oo )
9	8	adantl 275	. . . . . 6 |- ( ( B e. RR* /\ C e. RR* ) -> C <_ +oo )
10		pnfxr 7972	. . . . . . 7 |- +oo e. RR*
11		xrltletr 9764	. . . . . . 7 |- ( ( B e. RR* /\ C e. RR* /\ +oo e. RR* ) -> ( ( B < C /\ C <_ +oo ) -> B < +oo ) )
12	10, 11	mp3an3 1321	. . . . . 6 |- ( ( B e. RR* /\ C e. RR* ) -> ( ( B < C /\ C <_ +oo ) -> B < +oo ) )
13	9, 12	mpan2d 426	. . . . 5 |- ( ( B e. RR* /\ C e. RR* ) -> ( B < C -> B < +oo ) )
14	13	3adant1 1010	. . . 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 9776	. . . 4 |- ( B e. RR* -> ( B e. RR <-> ( -oo < B /\ B < +oo ) ) )
17	16	3ad2ant2 1014	. . 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 973   e. wcel 2141   class class class wbr 3989   RRcr 7773   +oocpnf 7951   -oocmnf 7952   RR*cxr 7953   < clt 7954   <_ cle 7955

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-po 4281  df-iso 4282  df-xp 4617  df-cnv 4619  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960

This theorem is referenced by:  elioore  9869  tgioo  13340