Theorem xrre 9756

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 9716	. . . . . 6 |- ( B e. RR -> B < +oo )
3	2	adantl 275	. . . . 5 |- ( ( A e. RR* /\ B e. RR ) -> B < +oo )
4		rexr 7944	. . . . . 6 |- ( B e. RR -> B e. RR* )
5		pnfxr 7951	. . . . . . 7 |- +oo e. RR*
6		xrlelttr 9742	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* /\ +oo e. RR* ) -> ( ( A <_ B /\ B < +oo ) -> A < +oo ) )
7	5, 6	mp3an3 1316	. . . . . 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 9755	. . 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 934	1 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( -oo < A /\ A <_ B ) ) -> A e. RR )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   e. wcel 2136   class class class wbr 3982   RRcr 7752   +oocpnf 7930   -oocmnf 7931   RR*cxr 7932   < clt 7933   <_ cle 7934

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867

This theorem depends on definitions:  df-bi 116  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-po 4274  df-iso 4275  df-xp 4610  df-cnv 4612  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939

This theorem is referenced by:  xrrege0  9761  pcgcd1  12259  tgioo  13186