Theorem xrre 8887

Description: A way of proving that an extended real is real. (Contributed by NM, 9-Mar-2006.)

Assertion

Ref	Expression
xrre	|- ( ( ( A e. RR* /\ B e. RR ) /\ ( -oo < A /\ A <_ B ) ) -> A e. RR )

Proof of Theorem xrre

Step	Hyp	Ref	Expression
1		simprl 497	. 2 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( -oo < A /\ A <_ B ) ) -> -oo < A )
2		ltpnf 8856	. . . . . 6 |- ( B e. RR -> B < +oo )
3	2	adantl 271	. . . . 5 |- ( ( A e. RR* /\ B e. RR ) -> B < +oo )
4		rexr 7164	. . . . . 6 |- ( B e. RR -> B e. RR* )
5		pnfxr 8846	. . . . . . 7 |- +oo e. RR*
6		xrlelttr 8876	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* /\ +oo e. RR* ) -> ( ( A <_ B /\ B < +oo ) -> A < +oo ) )
7	5, 6	mp3an3 1257	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* ) -> ( ( A <_ B /\ B < +oo ) -> A < +oo ) )
8	4, 7	sylan2 280	. . . . 5 |- ( ( A e. RR* /\ B e. RR ) -> ( ( A <_ B /\ B < +oo ) -> A < +oo ) )
9	3, 8	mpan2d 418	. . . 4 |- ( ( A e. RR* /\ B e. RR ) -> ( A <_ B -> A < +oo ) )
10	9	imp 122	. . 3 |- ( ( ( A e. RR* /\ B e. RR ) /\ A <_ B ) -> A < +oo )
11	10	adantrl 461	. 2 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( -oo < A /\ A <_ B ) ) -> A < +oo )
12		xrrebnd 8886	. . 3 |- ( A e. RR* -> ( A e. RR <-> ( -oo < A /\ A < +oo ) ) )
13	12	ad2antrr 471	. 2 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( -oo < A /\ A <_ B ) ) -> ( A e. RR <-> ( -oo < A /\ A < +oo ) ) )
14	1, 11, 13	mpbir2and 885	1 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( -oo < A /\ A <_ B ) ) -> A e. RR )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   e. wcel 1433   class class class wbr 3785   RRcr 6980   +oocpnf 7150   -oocmnf 7151   RR*cxr 7152   < clt 7153   <_ cle 7154

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 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-cnex 7067  ax-resscn 7068  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090

This theorem depends on definitions:  df-bi 115  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-rab 2357  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-po 4051  df-iso 4052  df-xp 4369  df-cnv 4371  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159

This theorem is referenced by:  xrrege0  8892