Theorem xrrebnd 9595

Description: An extended real is real iff it is strictly bounded by infinities. (Contributed by NM, 2-Feb-2006.)

Assertion

Ref	Expression
xrrebnd	|- ( A e. RR* -> ( A e. RR <-> ( -oo < A /\ A < +oo ) ) )

Proof of Theorem xrrebnd

Step	Hyp	Ref	Expression
1		elxr 9556	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		id 19	. . . 4 |- ( A e. RR -> A e. RR )
3		mnflt 9562	. . . . 5 |- ( A e. RR -> -oo < A )
4		ltpnf 9560	. . . . 5 |- ( A e. RR -> A < +oo )
5	3, 4	jca 304	. . . 4 |- ( A e. RR -> ( -oo < A /\ A < +oo ) )
6	2, 5	2thd 174	. . 3 |- ( A e. RR -> ( A e. RR <-> ( -oo < A /\ A < +oo ) ) )
7		renepnf 7806	. . . . 5 |- ( A e. RR -> A =/= +oo )
8	7	necon2bi 2361	. . . 4 |- ( A = +oo -> -. A e. RR )
9		pnfxr 7811	. . . . . . 7 |- +oo e. RR*
10		xrltnr 9559	. . . . . . 7 |- ( +oo e. RR* -> -. +oo < +oo )
11	9, 10	ax-mp 5	. . . . . 6 |- -. +oo < +oo
12		breq1 3927	. . . . . 6 |- ( A = +oo -> ( A < +oo <-> +oo < +oo ) )
13	11, 12	mtbiri 664	. . . . 5 |- ( A = +oo -> -. A < +oo )
14	13	intnand 916	. . . 4 |- ( A = +oo -> -. ( -oo < A /\ A < +oo ) )
15	8, 14	2falsed 691	. . 3 |- ( A = +oo -> ( A e. RR <-> ( -oo < A /\ A < +oo ) ) )
16		renemnf 7807	. . . . 5 |- ( A e. RR -> A =/= -oo )
17	16	necon2bi 2361	. . . 4 |- ( A = -oo -> -. A e. RR )
18		mnfxr 7815	. . . . . . 7 |- -oo e. RR*
19		xrltnr 9559	. . . . . . 7 |- ( -oo e. RR* -> -. -oo < -oo )
20	18, 19	ax-mp 5	. . . . . 6 |- -. -oo < -oo
21		breq2 3928	. . . . . 6 |- ( A = -oo -> ( -oo < A <-> -oo < -oo ) )
22	20, 21	mtbiri 664	. . . . 5 |- ( A = -oo -> -. -oo < A )
23	22	intnanrd 917	. . . 4 |- ( A = -oo -> -. ( -oo < A /\ A < +oo ) )
24	17, 23	2falsed 691	. . 3 |- ( A = -oo -> ( A e. RR <-> ( -oo < A /\ A < +oo ) ) )
25	6, 15, 24	3jaoi 1281	. 2 |- ( ( A e. RR \/ A = +oo \/ A = -oo ) -> ( A e. RR <-> ( -oo < A /\ A < +oo ) ) )
26	1, 25	sylbi 120	1 |- ( A e. RR* -> ( A e. RR <-> ( -oo < A /\ A < +oo ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ w3o 961   = wceq 1331   e. wcel 1480   class class class wbr 3924   RRcr 7612   +oocpnf 7790   -oocmnf 7791   RR*cxr 7792   < clt 7793

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-cnex 7704  ax-resscn 7705  ax-pre-ltirr 7725

This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-xp 4540  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798

This theorem is referenced by:  xrre  9596  xrre2  9597  xrre3  9598  elioc2  9712  elico2  9713  elicc2  9714  xblpnfps  12556  xblpnf  12557