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Theorem elioc2 12236
Description: Membership in an open-below, closed-above real interval. (Contributed by Paul Chapman, 30-Dec-2007.) (Revised by Mario Carneiro, 14-Jun-2014.)
Assertion
Ref Expression
elioc2 |- ( ( A e. RR* /\ B e. RR ) -> ( C e. ( A (,] B ) <-> ( C e. RR /\ A < C /\ C <_ B ) ) )
Proof of Theorem elioc2
StepHypRef Expression
1 rexr 10085 . . 3 |- ( B e. RR -> B e. RR* )
2 elioc1 12217 . . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( C e. ( A (,] B ) <-> ( C e. RR* /\ A < C /\ C <_ B ) ) )
31, 2sylan2 491 . 2 |- ( ( A e. RR* /\ B e. RR ) -> ( C e. ( A (,] B ) <-> ( C e. RR* /\ A < C /\ C <_ B ) ) )
4 mnfxr 10096 . . . . . . . 8 |- -oo e. RR*
54a1i 11 . . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( C e. RR* /\ A < C /\ C <_ B ) ) -> -oo e. RR* )
6 simpll 790 . . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( C e. RR* /\ A < C /\ C <_ B ) ) -> A e. RR* )
7 simpr1 1067 . . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( C e. RR* /\ A < C /\ C <_ B ) ) -> C e. RR* )
8 mnfle 11969 . . . . . . . 8 |- ( A e. RR* -> -oo <_ A )
98ad2antrr 762 . . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( C e. RR* /\ A < C /\ C <_ B ) ) -> -oo <_ A )
10 simpr2 1068 . . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( C e. RR* /\ A < C /\ C <_ B ) ) -> A < C )
115, 6, 7, 9, 10xrlelttrd 11991 . . . . . 6 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( C e. RR* /\ A < C /\ C <_ B ) ) -> -oo < C )
121ad2antlr 763 . . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( C e. RR* /\ A < C /\ C <_ B ) ) -> B e. RR* )
13 pnfxr 10092 . . . . . . . 8 |- +oo e. RR*
1413a1i 11 . . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( C e. RR* /\ A < C /\ C <_ B ) ) -> +oo e. RR* )
15 simpr3 1069 . . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( C e. RR* /\ A < C /\ C <_ B ) ) -> C <_ B )
16 ltpnf 11954 . . . . . . . 8 |- ( B e. RR -> B < +oo )
1716ad2antlr 763 . . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( C e. RR* /\ A < C /\ C <_ B ) ) -> B < +oo )
187, 12, 14, 15, 17xrlelttrd 11991 . . . . . 6 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( C e. RR* /\ A < C /\ C <_ B ) ) -> C < +oo )
19 xrrebnd 11999 . . . . . . 7 |- ( C e. RR* -> ( C e. RR <-> ( -oo < C /\ C < +oo ) ) )
207, 19syl 17 . . . . . 6 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( C e. RR* /\ A < C /\ C <_ B ) ) -> ( C e. RR <-> ( -oo < C /\ C < +oo ) ) )
2111, 18, 20mpbir2and 957 . . . . 5 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( C e. RR* /\ A < C /\ C <_ B ) ) -> C e. RR )
2221, 10, 153jca 1242 . . . 4 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( C e. RR* /\ A < C /\ C <_ B ) ) -> ( C e. RR /\ A < C /\ C <_ B ) )
2322ex 450 . . 3 |- ( ( A e. RR* /\ B e. RR ) -> ( ( C e. RR* /\ A < C /\ C <_ B ) -> ( C e. RR /\ A < C /\ C <_ B ) ) )
24 rexr 10085 . . . 4 |- ( C e. RR -> C e. RR* )
25243anim1i 1248 . . 3 |- ( ( C e. RR /\ A < C /\ C <_ B ) -> ( C e. RR* /\ A < C /\ C <_ B ) )
2623, 25impbid1 215 . 2 |- ( ( A e. RR* /\ B e. RR ) -> ( ( C e. RR* /\ A < C /\ C <_ B ) <-> ( C e. RR /\ A < C /\ C <_ B ) ) )
273, 26bitrd 268 1 |- ( ( A e. RR* /\ B e. RR ) -> ( C e. ( A (,] B ) <-> ( C e. RR /\ A < C /\ C <_ B ) ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   e. wcel 1990   class class class wbr 4653  (class class class)co 6650   RRcr 9935   +oocpnf 10071   -oocmnf 10072   RR*cxr 10073   < clt 10074   <_ cle 10075   (,]cioc 12176
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-ov 6653  df-oprab 6654  df-mpt2 6655  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  df-ioc 12180
This theorem is referenced by:  iocssre  12253  ef01bndlem  14914  sin01bnd  14915  cos01bnd  14916  cos1bnd  14917  sinltx  14919  sin01gt0  14920  cos01gt0  14921  sin02gt0  14922  sincos1sgn  14923  sincos2sgn  14924  icoopnst  22738  iocopnst  22739  ismbf3d  23421  aaliou3lem2  24098  aaliou3lem3  24099  pilem2  24206  sinhalfpilem  24215  sincosq1lem  24249  coseq0negpitopi  24255  tangtx  24257  sincos4thpi  24265  efif1olem1  24288  efif1olem2  24289  efif1o  24292  efifo  24293  ellogrn  24306  logimclad  24319  ellogdm  24385  logdmnrp  24387  dvloglem  24394  dvlog2lem  24398  asinneg  24613  atans2  24658  ressatans  24661  abvcxp  25304  ostth2  25326  xrge0iifcv  29980  xrge0iifiso  29981  xrge0iifhom  29983  sinccvglem  31566  bj-pinftyccb  33108  bj-pinftynminfty  33114  dvasin  33496  areacirclem4  33503  gtnelioc  39712  limcicciooub  39869  fourierdlem4  40328  fourierdlem26  40350  fourierdlem33  40357  fourierdlem37  40361  fourierdlem65  40388  fourierdlem79  40402  fouriersw  40448
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