Theorem xrrebnd 9819

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 9776	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		id 19	. . . 4 |- ( A e. RR -> A e. RR )
3		mnflt 9783	. . . . 5 |- ( A e. RR -> -oo < A )
4		ltpnf 9780	. . . . 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 8005	. . . . 5 |- ( A e. RR -> A =/= +oo )
8	7	necon2bi 2402	. . . 4 |- ( A = +oo -> -. A e. RR )
9		pnfxr 8010	. . . . . . 7 |- +oo e. RR*
10		xrltnr 9779	. . . . . . 7 |- ( +oo e. RR* -> -. +oo < +oo )
11	9, 10	ax-mp 5	. . . . . 6 |- -. +oo < +oo
12		breq1 4007	. . . . . 6 |- ( A = +oo -> ( A < +oo <-> +oo < +oo ) )
13	11, 12	mtbiri 675	. . . . 5 |- ( A = +oo -> -. A < +oo )
14	13	intnand 931	. . . 4 |- ( A = +oo -> -. ( -oo < A /\ A < +oo ) )
15	8, 14	2falsed 702	. . 3 |- ( A = +oo -> ( A e. RR <-> ( -oo < A /\ A < +oo ) ) )
16		renemnf 8006	. . . . 5 |- ( A e. RR -> A =/= -oo )
17	16	necon2bi 2402	. . . 4 |- ( A = -oo -> -. A e. RR )
18		mnfxr 8014	. . . . . . 7 |- -oo e. RR*
19		xrltnr 9779	. . . . . . 7 |- ( -oo e. RR* -> -. -oo < -oo )
20	18, 19	ax-mp 5	. . . . . 6 |- -. -oo < -oo
21		breq2 4008	. . . . . 6 |- ( A = -oo -> ( -oo < A <-> -oo < -oo ) )
22	20, 21	mtbiri 675	. . . . 5 |- ( A = -oo -> -. -oo < A )
23	22	intnanrd 932	. . . 4 |- ( A = -oo -> -. ( -oo < A /\ A < +oo ) )
24	17, 23	2falsed 702	. . 3 |- ( A = -oo -> ( A e. RR <-> ( -oo < A /\ A < +oo ) ) )
25	6, 15, 24	3jaoi 1303	. 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 977   = wceq 1353   e. wcel 2148   class class class wbr 4004   RRcr 7810   +oocpnf 7989   -oocmnf 7990   RR*cxr 7991   < clt 7992

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-cnex 7902  ax-resscn 7903  ax-pre-ltirr 7923

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2740  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-xp 4633  df-pnf 7994  df-mnf 7995  df-xr 7996  df-ltxr 7997

This theorem is referenced by:  xrre  9820  xrre2  9821  xrre3  9822  elioc2  9936  elico2  9937  elicc2  9938  xblpnfps  13901  xblpnf  13902