Theorem xrrebnd 10011

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 9968	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		id 19	. . . 4 |- ( A e. RR -> A e. RR )
3		mnflt 9975	. . . . 5 |- ( A e. RR -> -oo < A )
4		ltpnf 9972	. . . . 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 8190	. . . . 5 |- ( A e. RR -> A =/= +oo )
8	7	necon2bi 2455	. . . 4 |- ( A = +oo -> -. A e. RR )
9		pnfxr 8195	. . . . . . 7 |- +oo e. RR*
10		xrltnr 9971	. . . . . . 7 |- ( +oo e. RR* -> -. +oo < +oo )
11	9, 10	ax-mp 5	. . . . . 6 |- -. +oo < +oo
12		breq1 4085	. . . . . 6 |- ( A = +oo -> ( A < +oo <-> +oo < +oo ) )
13	11, 12	mtbiri 679	. . . . 5 |- ( A = +oo -> -. A < +oo )
14	13	intnand 936	. . . 4 |- ( A = +oo -> -. ( -oo < A /\ A < +oo ) )
15	8, 14	2falsed 707	. . 3 |- ( A = +oo -> ( A e. RR <-> ( -oo < A /\ A < +oo ) ) )
16		renemnf 8191	. . . . 5 |- ( A e. RR -> A =/= -oo )
17	16	necon2bi 2455	. . . 4 |- ( A = -oo -> -. A e. RR )
18		mnfxr 8199	. . . . . . 7 |- -oo e. RR*
19		xrltnr 9971	. . . . . . 7 |- ( -oo e. RR* -> -. -oo < -oo )
20	18, 19	ax-mp 5	. . . . . 6 |- -. -oo < -oo
21		breq2 4086	. . . . . 6 |- ( A = -oo -> ( -oo < A <-> -oo < -oo ) )
22	20, 21	mtbiri 679	. . . . 5 |- ( A = -oo -> -. -oo < A )
23	22	intnanrd 937	. . . 4 |- ( A = -oo -> -. ( -oo < A /\ A < +oo ) )
24	17, 23	2falsed 707	. . 3 |- ( A = -oo -> ( A e. RR <-> ( -oo < A /\ A < +oo ) ) )
25	6, 15, 24	3jaoi 1337	. 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 1001   = wceq 1395   e. wcel 2200   class class class wbr 4082   RRcr 7994   +oocpnf 8174   -oocmnf 8175   RR*cxr 8176   < clt 8177

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-cnex 8086  ax-resscn 8087  ax-pre-ltirr 8107

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-xp 4724  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182

This theorem is referenced by:  xrre  10012  xrre2  10013  xrre3  10014  elioc2  10128  elico2  10129  elicc2  10130  xblpnfps  15066  xblpnf  15067