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Theorem elioc2 9935
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 8002 . . 3 |- ( B e. RR -> B e. RR* )
2 elioc1 9921 . . 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 8013 . . . . . . . 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 1003 . . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( C e. RR* /\ A < C /\ C <_ B ) ) -> C e. RR* )
8 mnfle 9791 . . . . . . . 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 1004 . . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( C e. RR* /\ A < C /\ C <_ B ) ) -> A < C )
115, 6, 7, 9, 10xrlelttrd 9809 . . . . . 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 8009 . . . . . . . 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 1005 . . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( C e. RR* /\ A < C /\ C <_ B ) ) -> C <_ B )
16 ltpnf 9779 . . . . . . . 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 9809 . . . . . 6 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( C e. RR* /\ A < C /\ C <_ B ) ) -> C < +oo )
19 xrrebnd 9818 . . . . . . 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 944 . . . . 5 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( C e. RR* /\ A < C /\ C <_ B ) ) -> C e. RR )
2221, 10, 153jca 1177 . . . 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 8002 . . . 4 |- ( C e. RR -> C e. RR* )
25243anim1i 1185 . . 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 978   e. wcel 2148   class class class wbr 4003  (class class class)co 5874   RRcr 7809   +oocpnf 7988   -oocmnf 7989   RR*cxr 7990   < clt 7991   <_ cle 7992   (,]cioc 9888
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-id 4293  df-po 4296  df-iso 4297  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-iota 5178  df-fun 5218  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-pnf 7993  df-mnf 7994  df-xr 7995  df-ltxr 7996  df-le 7997  df-ioc 9892
This theorem is referenced by:  iocssre  9952  ef01bndlem  11763  sin01bnd  11764  cos01bnd  11765  cos1bnd  11766  sin01gt0  11768  cos01gt0  11769  sin02gt0  11770  sincos1sgn  11771  sincos2sgn  11772  cos12dec  11774  sin0pilem1  14172  sin0pilem2  14173  sinhalfpilem  14182  sincosq1lem  14216  coseq0negpitopi  14227  tangtx  14229  sincos4thpi  14231  pigt3  14235
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