Theorem xrrebnd 11999

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		mnflt 11957	. . 3 |- ( A e. RR -> -oo < A )
2		ltpnf 11954	. . 3 |- ( A e. RR -> A < +oo )
3	1, 2	jca 554	. 2 |- ( A e. RR -> ( -oo < A /\ A < +oo ) )
4		nltpnft 11995	. . . . . 6 |- ( A e. RR* -> ( A = +oo <-> -. A < +oo ) )
5		ngtmnft 11997	. . . . . 6 |- ( A e. RR* -> ( A = -oo <-> -. -oo < A ) )
6	4, 5	orbi12d 746	. . . . 5 |- ( A e. RR* -> ( ( A = +oo \/ A = -oo ) <-> ( -. A < +oo \/ -. -oo < A ) ) )
7		ianor 509	. . . . . 6 |- ( -. ( -oo < A /\ A < +oo ) <-> ( -. -oo < A \/ -. A < +oo ) )
8		orcom 402	. . . . . 6 |- ( ( -. -oo < A \/ -. A < +oo ) <-> ( -. A < +oo \/ -. -oo < A ) )
9	7, 8	bitr2i 265	. . . . 5 |- ( ( -. A < +oo \/ -. -oo < A ) <-> -. ( -oo < A /\ A < +oo ) )
10	6, 9	syl6bb 276	. . . 4 |- ( A e. RR* -> ( ( A = +oo \/ A = -oo ) <-> -. ( -oo < A /\ A < +oo ) ) )
11	10	con2bid 344	. . 3 |- ( A e. RR* -> ( ( -oo < A /\ A < +oo ) <-> -. ( A = +oo \/ A = -oo ) ) )
12		elxr 11950	. . . . 5 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
13		3orass 1040	. . . . . 6 |- ( ( A e. RR \/ A = +oo \/ A = -oo ) <-> ( A e. RR \/ ( A = +oo \/ A = -oo ) ) )
14		orcom 402	. . . . . 6 |- ( ( A e. RR \/ ( A = +oo \/ A = -oo ) ) <-> ( ( A = +oo \/ A = -oo ) \/ A e. RR ) )
15	13, 14	bitri 264	. . . . 5 |- ( ( A e. RR \/ A = +oo \/ A = -oo ) <-> ( ( A = +oo \/ A = -oo ) \/ A e. RR ) )
16	12, 15	sylbb 209	. . . 4 |- ( A e. RR* -> ( ( A = +oo \/ A = -oo ) \/ A e. RR ) )
17	16	ord 392	. . 3 |- ( A e. RR* -> ( -. ( A = +oo \/ A = -oo ) -> A e. RR ) )
18	11, 17	sylbid 230	. 2 |- ( A e. RR* -> ( ( -oo < A /\ A < +oo ) -> A e. RR ) )
19	3, 18	impbid2 216	1 |- ( A e. RR* -> ( A e. RR <-> ( -oo < A /\ A < +oo ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   \/ w3o 1036   = wceq 1483   e. wcel 1990   class class class wbr 4653   RRcr 9935   +oocpnf 10071   -oocmnf 10072   RR*cxr 10073   < clt 10074

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-pre-lttri 10010  ax-pre-lttrn 10011

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080

This theorem is referenced by:  xrre  12000  xrre2  12001  xrre3  12002  supxrre1  12160  elioc2  12236  elico2  12237  elicc2  12238  xblpnfps  22200  xblpnf  22201  isnghm3  22529  ovoliun  23273  ovolicopnf  23292  voliunlem3  23320  volsup  23324  itg2seq  23509  nmblore  27641  nmopre  28729  supxrgere  39549  supxrgelem  39553  supxrge  39554  suplesup  39555  infrpge  39567  limsupre  39873