Theorem xrrebnd 9833

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 9790	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		id 19	. . . 4 |- ( A e. RR -> A e. RR )
3		mnflt 9797	. . . . 5 |- ( A e. RR -> -oo < A )
4		ltpnf 9794	. . . . 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 8019	. . . . 5 |- ( A e. RR -> A =/= +oo )
8	7	necon2bi 2412	. . . 4 |- ( A = +oo -> -. A e. RR )
9		pnfxr 8024	. . . . . . 7 |- +oo e. RR*
10		xrltnr 9793	. . . . . . 7 |- ( +oo e. RR* -> -. +oo < +oo )
11	9, 10	ax-mp 5	. . . . . 6 |- -. +oo < +oo
12		breq1 4018	. . . . . 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 8020	. . . . 5 |- ( A e. RR -> A =/= -oo )
17	16	necon2bi 2412	. . . 4 |- ( A = -oo -> -. A e. RR )
18		mnfxr 8028	. . . . . . 7 |- -oo e. RR*
19		xrltnr 9793	. . . . . . 7 |- ( -oo e. RR* -> -. -oo < -oo )
20	18, 19	ax-mp 5	. . . . . 6 |- -. -oo < -oo
21		breq2 4019	. . . . . 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 1313	. 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 978   = wceq 1363   e. wcel 2158   class class class wbr 4015   RRcr 7824   +oocpnf 8003   -oocmnf 8004   RR*cxr 8005   < clt 8006

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7916  ax-resscn 7917  ax-pre-ltirr 7937

This theorem depends on definitions:  df-bi 117  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-xp 4644  df-pnf 8008  df-mnf 8009  df-xr 8010  df-ltxr 8011

This theorem is referenced by:  xrre  9834  xrre2  9835  xrre3  9836  elioc2  9950  elico2  9951  elicc2  9952  xblpnfps  14194  xblpnf  14195