Theorem elico2 10930

Description: Membership in a closed-below, open-above real interval. (Contributed by Paul Chapman, 21-Jan-2008.) (Revised by Mario Carneiro, 14-Jun-2014.)

Assertion

Ref	Expression
elico2	|- ( ( A e. RR /\ B e. RR* ) -> ( C e. ( A [,) B ) <-> ( C e. RR /\ A <_ C /\ C < B ) ) )

Proof of Theorem elico2

Step	Hyp	Ref	Expression
1		rexr 9086	. . 3 |- ( A e. RR -> A e. RR* )
2		elico1 10915	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( C e. ( A [,) B ) <-> ( C e. RR* /\ A <_ C /\ C < B ) ) )
3	1, 2	sylan 458	. 2 |- ( ( A e. RR /\ B e. RR* ) -> ( C e. ( A [,) B ) <-> ( C e. RR* /\ A <_ C /\ C < B ) ) )
4		mnfxr 10670	. . . . . . . 8 |- -oo e. RR*
5	4	a1i 11	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C < B ) ) -> -oo e. RR* )
6	1	ad2antrr 707	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C < B ) ) -> A e. RR* )
7		simpr1 963	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C < B ) ) -> C e. RR* )
8		mnflt 10678	. . . . . . . 8 |- ( A e. RR -> -oo < A )
9	8	ad2antrr 707	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C < B ) ) -> -oo < A )
10		simpr2 964	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C < B ) ) -> A <_ C )
11	5, 6, 7, 9, 10	xrltletrd 10707	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C < B ) ) -> -oo < C )
12		simplr 732	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C < B ) ) -> B e. RR* )
13		pnfxr 10669	. . . . . . . 8 |- +oo e. RR*
14	13	a1i 11	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C < B ) ) -> +oo e. RR* )
15		simpr3 965	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C < B ) ) -> C < B )
16		pnfge 10683	. . . . . . . 8 |- ( B e. RR* -> B <_ +oo )
17	16	ad2antlr 708	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C < B ) ) -> B <_ +oo )
18	7, 12, 14, 15, 17	xrltletrd 10707	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C < B ) ) -> C < +oo )
19		xrrebnd 10712	. . . . . . 7 |- ( C e. RR* -> ( C e. RR <-> ( -oo < C /\ C < +oo ) ) )
20	7, 19	syl 16	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C < B ) ) -> ( C e. RR <-> ( -oo < C /\ C < +oo ) ) )
21	11, 18, 20	mpbir2and 889	. . . . 5 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C < B ) ) -> C e. RR )
22	21, 10, 15	3jca 1134	. . . 4 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C < B ) ) -> ( C e. RR /\ A <_ C /\ C < B ) )
23	22	ex 424	. . 3 |- ( ( A e. RR /\ B e. RR* ) -> ( ( C e. RR* /\ A <_ C /\ C < B ) -> ( C e. RR /\ A <_ C /\ C < B ) ) )
24		rexr 9086	. . . 4 |- ( C e. RR -> C e. RR* )
25	24	3anim1i 1140	. . 3 |- ( ( C e. RR /\ A <_ C /\ C < B ) -> ( C e. RR* /\ A <_ C /\ C < B ) )
26	23, 25	impbid1 195	. 2 |- ( ( A e. RR /\ B e. RR* ) -> ( ( C e. RR* /\ A <_ C /\ C < B ) <-> ( C e. RR /\ A <_ C /\ C < B ) ) )
27	3, 26	bitrd 245	1 |- ( ( A e. RR /\ B e. RR* ) -> ( C e. ( A [,) B ) <-> ( C e. RR /\ A <_ C /\ C < B ) ) )

Syntax hints:   -> wi 4   <-> wb 177   /\ wa 359   /\ w3a 936   e. wcel 1721   class class class wbr 4172  (class class class)co 6040   RRcr 8945   +oocpnf 9073   -oocmnf 9074   RR*cxr 9075   < clt 9076   <_ cle 9077   [,)cico 10874

This theorem is referenced by:  icossre  10947  elicopnf  10956  icoshft  10975  metustexhalfOLD  18546  metustexhalf  18547  cnbl0  18761  icoopnst  18917  iocopnst  18918  icopnfcnv  18920  icopnfhmeo  18921  iccpnfcnv  18922  psercnlem2  20293  psercnlem1  20294  psercn  20295  abelth  20310  tanord1  20392  tanord  20393  efopnlem1  20500  logtayl  20504  rlimcnp  20757  rlimcnp2  20758  dchrvmasumlem2  21145  dchrvmasumiflem1  21148  pntlemb  21244  pnt  21261  ubico  24091  voliune  24538  volfiniune  24539  dya2icoseg  24580  sibfof  24607  itg2addnclem2  26156  icodiamlt  26773  modelico  26774

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-pre-lttri 9020  ax-pre-lttrn 9021

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-ico 10878