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Theorem elioc2 9822
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 7906 . . 3 |- ( B e. RR -> B e. RR* )
2 elioc1 9808 . . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( C e. ( A (,] B ) <-> ( C e. RR* /\ A < C /\ C <_ B ) ) )
31, 2sylan2 284 . 2 |- ( ( A e. RR* /\ B e. RR ) -> ( C e. ( A (,] B ) <-> ( C e. RR* /\ A < C /\ C <_ B ) ) )
4 mnfxr 7917 . . . . . . . 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 519 . . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( C e. RR* /\ A < C /\ C <_ B ) ) -> A e. RR* )
7 simpr1 988 . . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( C e. RR* /\ A < C /\ C <_ B ) ) -> C e. RR* )
8 mnfle 9681 . . . . . . . 8 |- ( A e. RR* -> -oo <_ A )
98ad2antrr 480 . . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( C e. RR* /\ A < C /\ C <_ B ) ) -> -oo <_ A )
10 simpr2 989 . . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( C e. RR* /\ A < C /\ C <_ B ) ) -> A < C )
115, 6, 7, 9, 10xrlelttrd 9696 . . . . . 6 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( C e. RR* /\ A < C /\ C <_ B ) ) -> -oo < C )
121ad2antlr 481 . . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( C e. RR* /\ A < C /\ C <_ B ) ) -> B e. RR* )
13 pnfxr 7913 . . . . . . . 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 990 . . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( C e. RR* /\ A < C /\ C <_ B ) ) -> C <_ B )
16 ltpnf 9669 . . . . . . . 8 |- ( B e. RR -> B < +oo )
1716ad2antlr 481 . . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( C e. RR* /\ A < C /\ C <_ B ) ) -> B < +oo )
187, 12, 14, 15, 17xrlelttrd 9696 . . . . . 6 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( C e. RR* /\ A < C /\ C <_ B ) ) -> C < +oo )
19 xrrebnd 9705 . . . . . . 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 929 . . . . 5 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( C e. RR* /\ A < C /\ C <_ B ) ) -> C e. RR )
2221, 10, 153jca 1162 . . . 4 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( C e. RR* /\ A < C /\ C <_ B ) ) -> ( C e. RR /\ A < C /\ C <_ B ) )
2322ex 114 . . 3 |- ( ( A e. RR* /\ B e. RR ) -> ( ( C e. RR* /\ A < C /\ C <_ B ) -> ( C e. RR /\ A < C /\ C <_ B ) ) )
24 rexr 7906 . . . 4 |- ( C e. RR -> C e. RR* )
25243anim1i 1168 . . 3 |- ( ( C e. RR /\ A < C /\ C <_ B ) -> ( C e. RR* /\ A < C /\ C <_ B ) )
2623, 25impbid1 141 . 2 |- ( ( A e. RR* /\ B e. RR ) -> ( ( C e. RR* /\ A < C /\ C <_ B ) <-> ( C e. RR /\ A < C /\ C <_ B ) ) )
273, 26bitrd 187 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 103   <-> wb 104   /\ w3a 963   e. wcel 2128   class class class wbr 3965  (class class class)co 5818   RRcr 7714   +oocpnf 7892   -oocmnf 7893   RR*cxr 7894   < clt 7895   <_ cle 7896   (,]cioc 9775
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4134  ax-pr 4168  ax-un 4392  ax-setind 4494  ax-cnex 7806  ax-resscn 7807  ax-pre-ltirr 7827  ax-pre-ltwlin 7828  ax-pre-lttrn 7829
This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-id 4252  df-po 4255  df-iso 4256  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-iota 5132  df-fun 5169  df-fv 5175  df-ov 5821  df-oprab 5822  df-mpo 5823  df-pnf 7897  df-mnf 7898  df-xr 7899  df-ltxr 7900  df-le 7901  df-ioc 9779
This theorem is referenced by:  iocssre  9839  ef01bndlem  11635  sin01bnd  11636  cos01bnd  11637  cos1bnd  11638  sin01gt0  11640  cos01gt0  11641  sin02gt0  11642  sincos1sgn  11643  sincos2sgn  11644  cos12dec  11646  sin0pilem1  13062  sin0pilem2  13063  sinhalfpilem  13072  sincosq1lem  13106  coseq0negpitopi  13117  tangtx  13119  sincos4thpi  13121  pigt3  13125
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