Theorem xrrebnd 9911

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 9868	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		id 19	. . . 4 |- ( A e. RR -> A e. RR )
3		mnflt 9875	. . . . 5 |- ( A e. RR -> -oo < A )
4		ltpnf 9872	. . . . 5 |- ( A e. RR -> A < +oo )
5	3, 4	jca 306	. . . 4 |- ( A e. RR -> ( -oo < A /\ A < +oo ) )
6	2, 5	2thd 175	. . 3 |- ( A e. RR -> ( A e. RR <-> ( -oo < A /\ A < +oo ) ) )
7		renepnf 8091	. . . . 5 |- ( A e. RR -> A =/= +oo )
8	7	necon2bi 2422	. . . 4 |- ( A = +oo -> -. A e. RR )
9		pnfxr 8096	. . . . . . 7 |- +oo e. RR*
10		xrltnr 9871	. . . . . . 7 |- ( +oo e. RR* -> -. +oo < +oo )
11	9, 10	ax-mp 5	. . . . . 6 |- -. +oo < +oo
12		breq1 4037	. . . . . 6 |- ( A = +oo -> ( A < +oo <-> +oo < +oo ) )
13	11, 12	mtbiri 676	. . . . 5 |- ( A = +oo -> -. A < +oo )
14	13	intnand 932	. . . 4 |- ( A = +oo -> -. ( -oo < A /\ A < +oo ) )
15	8, 14	2falsed 703	. . 3 |- ( A = +oo -> ( A e. RR <-> ( -oo < A /\ A < +oo ) ) )
16		renemnf 8092	. . . . 5 |- ( A e. RR -> A =/= -oo )
17	16	necon2bi 2422	. . . 4 |- ( A = -oo -> -. A e. RR )
18		mnfxr 8100	. . . . . . 7 |- -oo e. RR*
19		xrltnr 9871	. . . . . . 7 |- ( -oo e. RR* -> -. -oo < -oo )
20	18, 19	ax-mp 5	. . . . . 6 |- -. -oo < -oo
21		breq2 4038	. . . . . 6 |- ( A = -oo -> ( -oo < A <-> -oo < -oo ) )
22	20, 21	mtbiri 676	. . . . 5 |- ( A = -oo -> -. -oo < A )
23	22	intnanrd 933	. . . 4 |- ( A = -oo -> -. ( -oo < A /\ A < +oo ) )
24	17, 23	2falsed 703	. . 3 |- ( A = -oo -> ( A e. RR <-> ( -oo < A /\ A < +oo ) ) )
25	6, 15, 24	3jaoi 1314	. 2 |- ( ( A e. RR \/ A = +oo \/ A = -oo ) -> ( A e. RR <-> ( -oo < A /\ A < +oo ) ) )
26	1, 25	sylbi 121	1 |- ( A e. RR* -> ( A e. RR <-> ( -oo < A /\ A < +oo ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ w3o 979   = wceq 1364   e. wcel 2167   class class class wbr 4034   RRcr 7895   +oocpnf 8075   -oocmnf 8076   RR*cxr 8077   < clt 8078

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-pre-ltirr 8008

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-xp 4670  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083

This theorem is referenced by:  xrre  9912  xrre2  9913  xrre3  9914  elioc2  10028  elico2  10029  elicc2  10030  xblpnfps  14718  xblpnf  14719