Theorem xrre 10222

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 535	. 2 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( -oo < A /\ A <_ B ) ) -> -oo < A )
2		ltpnf 10182	. . . . . 6 |- ( B e. RR -> B < +oo )
3	2	adantl 277	. . . . 5 |- ( ( A e. RR* /\ B e. RR ) -> B < +oo )
4		rexr 8371	. . . . . 6 |- ( B e. RR -> B e. RR* )
5		pnfxr 8378	. . . . . . 7 |- +oo e. RR*
6		xrlelttr 10208	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* /\ +oo e. RR* ) -> ( ( A <_ B /\ B < +oo ) -> A < +oo ) )
7	5, 6	mp3an3 1367	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* ) -> ( ( A <_ B /\ B < +oo ) -> A < +oo ) )
8	4, 7	sylan2 286	. . . . 5 |- ( ( A e. RR* /\ B e. RR ) -> ( ( A <_ B /\ B < +oo ) -> A < +oo ) )
9	3, 8	mpan2d 432	. . . 4 |- ( ( A e. RR* /\ B e. RR ) -> ( A <_ B -> A < +oo ) )
10	9	imp 124	. . 3 |- ( ( ( A e. RR* /\ B e. RR ) /\ A <_ B ) -> A < +oo )
11	10	adantrl 482	. 2 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( -oo < A /\ A <_ B ) ) -> A < +oo )
12		xrrebnd 10221	. . 3 |- ( A e. RR* -> ( A e. RR <-> ( -oo < A /\ A < +oo ) ) )
13	12	ad2antrr 492	. 2 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( -oo < A /\ A <_ B ) ) -> ( A e. RR <-> ( -oo < A /\ A < +oo ) ) )
14	1, 11, 13	mpbir2and 957	1 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( -oo < A /\ A <_ B ) ) -> A e. RR )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2209   class class class wbr 4130   RRcr 8178   +oocpnf 8357   -oocmnf 8358   RR*cxr 8359   < clt 8360   <_ cle 8361

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-po 4441  df-iso 4442  df-xp 4780  df-cnv 4782  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366

This theorem is used by:  xrrege0  10227  pcgcd1  13107  tgioo  15655