Theorem xrre 9724

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 521	. 2 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( -oo < A /\ A <_ B ) ) -> -oo < A )
2		ltpnf 9687	. . . . . 6 |- ( B e. RR -> B < +oo )
3	2	adantl 275	. . . . 5 |- ( ( A e. RR* /\ B e. RR ) -> B < +oo )
4		rexr 7923	. . . . . 6 |- ( B e. RR -> B e. RR* )
5		pnfxr 7930	. . . . . . 7 |- +oo e. RR*
6		xrlelttr 9710	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* /\ +oo e. RR* ) -> ( ( A <_ B /\ B < +oo ) -> A < +oo ) )
7	5, 6	mp3an3 1308	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* ) -> ( ( A <_ B /\ B < +oo ) -> A < +oo ) )
8	4, 7	sylan2 284	. . . . 5 |- ( ( A e. RR* /\ B e. RR ) -> ( ( A <_ B /\ B < +oo ) -> A < +oo ) )
9	3, 8	mpan2d 425	. . . 4 |- ( ( A e. RR* /\ B e. RR ) -> ( A <_ B -> A < +oo ) )
10	9	imp 123	. . 3 |- ( ( ( A e. RR* /\ B e. RR ) /\ A <_ B ) -> A < +oo )
11	10	adantrl 470	. 2 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( -oo < A /\ A <_ B ) ) -> A < +oo )
12		xrrebnd 9723	. . 3 |- ( A e. RR* -> ( A e. RR <-> ( -oo < A /\ A < +oo ) ) )
13	12	ad2antrr 480	. 2 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( -oo < A /\ A <_ B ) ) -> ( A e. RR <-> ( -oo < A /\ A < +oo ) ) )
14	1, 11, 13	mpbir2and 929	1 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( -oo < A /\ A <_ B ) ) -> A e. RR )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   e. wcel 2128   class class class wbr 3965   RRcr 7731   +oocpnf 7909   -oocmnf 7910   RR*cxr 7911   < clt 7912   <_ cle 7913

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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-cnex 7823  ax-resscn 7824  ax-pre-ltirr 7844  ax-pre-ltwlin 7845  ax-pre-lttrn 7846

This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-po 4256  df-iso 4257  df-xp 4592  df-cnv 4594  df-pnf 7914  df-mnf 7915  df-xr 7916  df-ltxr 7917  df-le 7918

This theorem is referenced by:  xrrege0  9729  tgioo  12957