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Theorem elioc2 10002
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 8065 . . 3 |- ( B e. RR -> B e. RR* )
2 elioc1 9988 . . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( C e. ( A (,] B ) <-> ( C e. RR* /\ A < C /\ C <_ B ) ) )
31, 2sylan2 286 . 2 |- ( ( A e. RR* /\ B e. RR ) -> ( C e. ( A (,] B ) <-> ( C e. RR* /\ A < C /\ C <_ B ) ) )
4 mnfxr 8076 . . . . . . . 8 |- -oo e. RR*
54a1i 9 . . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( C e. RR* /\ A < C /\ C <_ B ) ) -> -oo e. RR* )
6 simpll 527 . . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( C e. RR* /\ A < C /\ C <_ B ) ) -> A e. RR* )
7 simpr1 1005 . . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( C e. RR* /\ A < C /\ C <_ B ) ) -> C e. RR* )
8 mnfle 9858 . . . . . . . 8 |- ( A e. RR* -> -oo <_ A )
98ad2antrr 488 . . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( C e. RR* /\ A < C /\ C <_ B ) ) -> -oo <_ A )
10 simpr2 1006 . . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( C e. RR* /\ A < C /\ C <_ B ) ) -> A < C )
115, 6, 7, 9, 10xrlelttrd 9876 . . . . . 6 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( C e. RR* /\ A < C /\ C <_ B ) ) -> -oo < C )
121ad2antlr 489 . . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( C e. RR* /\ A < C /\ C <_ B ) ) -> B e. RR* )
13 pnfxr 8072 . . . . . . . 8 |- +oo e. RR*
1413a1i 9 . . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( C e. RR* /\ A < C /\ C <_ B ) ) -> +oo e. RR* )
15 simpr3 1007 . . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( C e. RR* /\ A < C /\ C <_ B ) ) -> C <_ B )
16 ltpnf 9846 . . . . . . . 8 |- ( B e. RR -> B < +oo )
1716ad2antlr 489 . . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( C e. RR* /\ A < C /\ C <_ B ) ) -> B < +oo )
187, 12, 14, 15, 17xrlelttrd 9876 . . . . . 6 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( C e. RR* /\ A < C /\ C <_ B ) ) -> C < +oo )
19 xrrebnd 9885 . . . . . . 7 |- ( C e. RR* -> ( C e. RR <-> ( -oo < C /\ C < +oo ) ) )
207, 19syl 14 . . . . . 6 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( C e. RR* /\ A < C /\ C <_ B ) ) -> ( C e. RR <-> ( -oo < C /\ C < +oo ) ) )
2111, 18, 20mpbir2and 946 . . . . 5 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( C e. RR* /\ A < C /\ C <_ B ) ) -> C e. RR )
2221, 10, 153jca 1179 . . . 4 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( C e. RR* /\ A < C /\ C <_ B ) ) -> ( C e. RR /\ A < C /\ C <_ B ) )
2322ex 115 . . 3 |- ( ( A e. RR* /\ B e. RR ) -> ( ( C e. RR* /\ A < C /\ C <_ B ) -> ( C e. RR /\ A < C /\ C <_ B ) ) )
24 rexr 8065 . . . 4 |- ( C e. RR -> C e. RR* )
25243anim1i 1187 . . 3 |- ( ( C e. RR /\ A < C /\ C <_ B ) -> ( C e. RR* /\ A < C /\ C <_ B ) )
2623, 25impbid1 142 . 2 |- ( ( A e. RR* /\ B e. RR ) -> ( ( C e. RR* /\ A < C /\ C <_ B ) <-> ( C e. RR /\ A < C /\ C <_ B ) ) )
273, 26bitrd 188 1 |- ( ( A e. RR* /\ B e. RR ) -> ( C e. ( A (,] B ) <-> ( C e. RR /\ A < C /\ C <_ B ) ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   e. wcel 2164   class class class wbr 4029  (class class class)co 5918   RRcr 7871   +oocpnf 8051   -oocmnf 8052   RR*cxr 8053   < clt 8054   <_ cle 8055   (,]cioc 9955
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-id 4324  df-po 4327  df-iso 4328  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-iota 5215  df-fun 5256  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-ioc 9959
This theorem is referenced by:  iocssre  10019  ef01bndlem  11899  sin01bnd  11900  cos01bnd  11901  cos1bnd  11902  sinltxirr  11904  sin01gt0  11905  cos01gt0  11906  sin02gt0  11907  sincos1sgn  11908  sincos2sgn  11909  cos12dec  11911  sin0pilem1  14916  sin0pilem2  14917  sinhalfpilem  14926  sincosq1lem  14960  coseq0negpitopi  14971  tangtx  14973  sincos4thpi  14975  pigt3  14979
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