Theorem elico2 11687

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 9675	. . 3 |- ( A e. RR -> A e. RR* )
2		elico1 11668	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( C e. ( A [,) B ) <-> ( C e. RR* /\ A <_ C /\ C < B ) ) )
3	1, 2	sylan 473	. 2 |- ( ( A e. RR /\ B e. RR* ) -> ( C e. ( A [,) B ) <-> ( C e. RR* /\ A <_ C /\ C < B ) ) )
4		mnfxr 11403	. . . . . . . 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 730	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C < B ) ) -> A e. RR* )
7		simpr1 1011	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C < B ) ) -> C e. RR* )
8		mnflt 11414	. . . . . . . 8 |- ( A e. RR -> -oo < A )
9	8	ad2antrr 730	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C < B ) ) -> -oo < A )
10		simpr2 1012	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C < B ) ) -> A <_ C )
11	5, 6, 7, 9, 10	xrltletrd 11447	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C < B ) ) -> -oo < C )
12		simplr 760	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C < B ) ) -> B e. RR* )
13		pnfxr 11401	. . . . . . . 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 1013	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C < B ) ) -> C < B )
16		pnfge 11421	. . . . . . . 8 |- ( B e. RR* -> B <_ +oo )
17	16	ad2antlr 731	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C < B ) ) -> B <_ +oo )
18	7, 12, 14, 15, 17	xrltletrd 11447	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C < B ) ) -> C < +oo )
19		xrrebnd 11452	. . . . . . 7 |- ( C e. RR* -> ( C e. RR <-> ( -oo < C /\ C < +oo ) ) )
20	7, 19	syl 17	. . . . . 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 930	. . . . 5 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C < B ) ) -> C e. RR )
22	21, 10, 15	3jca 1185	. . . 4 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C < B ) ) -> ( C e. RR /\ A <_ C /\ C < B ) )
23	22	ex 435	. . 3 |- ( ( A e. RR /\ B e. RR* ) -> ( ( C e. RR* /\ A <_ C /\ C < B ) -> ( C e. RR /\ A <_ C /\ C < B ) ) )
24		rexr 9675	. . . 4 |- ( C e. RR -> C e. RR* )
25	24	3anim1i 1191	. . 3 |- ( ( C e. RR /\ A <_ C /\ C < B ) -> ( C e. RR* /\ A <_ C /\ C < B ) )
26	23, 25	impbid1 206	. 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 256	1 |- ( ( A e. RR /\ B e. RR* ) -> ( C e. ( A [,) B ) <-> ( C e. RR /\ A <_ C /\ C < B ) ) )

Syntax hints:   -> wi 4   <-> wb 187   /\ wa 370   /\ w3a 982   e. wcel 1867   class class class wbr 4417  (class class class)co 6296   RRcr 9527   +oocpnf 9661   -oocmnf 9662   RR*cxr 9663   < clt 9664   <_ cle 9665   [,)cico 11626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-cnex 9584  ax-resscn 9585  ax-pre-lttri 9602  ax-pre-lttrn 9603

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-po 4766  df-so 4767  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-er 7362  df-en 7569  df-dom 7570  df-sdom 7571  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-ico 11630

This theorem is referenced by:  icossre  11704  elicopnf  11719  icoshft  11741  muladdmodid  12123  fprodge0  14014  fprodge1  14016  rge0srg  18965  metustexhalf  21495  cnbl0  21698  icoopnst  21856  iocopnst  21857  icopnfcnv  21859  icopnfhmeo  21860  iccpnfcnv  21861  psercnlem2  23241  psercnlem1  23242  psercn  23243  abelth  23258  tanord1  23348  tanord  23349  efopnlem1  23463  logtayl  23467  rlimcnp  23753  rlimcnp2  23754  dchrvmasumlem2  24196  dchrvmasumiflem1  24199  pntlemb  24295  pnt  24312  ubico  28190  xrge0slmod  28443  voliune  28888  volfiniune  28889  dya2icoseg  28935  sibfinima  28997  relowlpssretop  31498  tan2h  31641  itg2addnclem2  31698  icodiamlt  35374  modelico  35375  binomcxplemdvbinom  36343  binomcxplemcvg  36344  binomcxplemnotnn0  36346  limciccioolb  37277  fourierdlem32  37574  fourierdlem43  37585  fourierdlem63  37605  fourierdlem79  37621  fouriersw  37667  expnegico01  39088  dignnld  39188