Theorem xrre2 10046

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 10017	. . . . . . 7 |- ( A e. RR* -> -oo <_ A )
2	1	adantr 276	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* ) -> -oo <_ A )
3		mnfxr 8226	. . . . . . 7 |- -oo e. RR*
4		xrlelttr 10031	. . . . . . 7 |- ( ( -oo e. RR* /\ A e. RR* /\ B e. RR* ) -> ( ( -oo <_ A /\ A < B ) -> -oo < B ) )
5	3, 4	mp3an1 1358	. . . . . 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 1041	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A < B -> -oo < B ) )
8		pnfge 10014	. . . . . . 7 |- ( C e. RR* -> C <_ +oo )
9	8	adantl 277	. . . . . 6 |- ( ( B e. RR* /\ C e. RR* ) -> C <_ +oo )
10		pnfxr 8222	. . . . . . 7 |- +oo e. RR*
11		xrltletr 10032	. . . . . . 7 |- ( ( B e. RR* /\ C e. RR* /\ +oo e. RR* ) -> ( ( B < C /\ C <_ +oo ) -> B < +oo ) )
12	10, 11	mp3an3 1360	. . . . . 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 1039	. . . 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 10044	. . . 4 |- ( B e. RR* -> ( B e. RR <-> ( -oo < B /\ B < +oo ) ) )
17	16	3ad2ant2 1043	. . 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 1002   e. wcel 2200   class class class wbr 4086   RRcr 8021   +oocpnf 8201   -oocmnf 8202   RR*cxr 8203   < clt 8204   <_ cle 8205

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2802  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-po 4391  df-iso 4392  df-xp 4729  df-cnv 4731  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210

This theorem is referenced by:  elioore  10137  tgioo  15268