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Theorem elico2 9290
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
StepHypRef Expression
1 rexr 7480 . . 3  |-  ( A  e.  RR  ->  A  e.  RR* )
2 elico1 9276 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A [,) B )  <->  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B
) ) )
31, 2sylan 277 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( C  e.  ( A [,) B )  <-> 
( C  e.  RR*  /\  A  <_  C  /\  C  <  B ) ) )
4 mnfxr 7491 . . . . . . . 8  |- -oo  e.  RR*
54a1i 9 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B ) )  -> -oo  e.  RR* )
61ad2antrr 472 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B ) )  ->  A  e.  RR* )
7 simpr1 947 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B ) )  ->  C  e.  RR* )
8 mnflt 9188 . . . . . . . 8  |-  ( A  e.  RR  -> -oo  <  A )
98ad2antrr 472 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B ) )  -> -oo  <  A )
10 simpr2 948 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B ) )  ->  A  <_  C
)
115, 6, 7, 9, 10xrltletrd 9211 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B ) )  -> -oo  <  C )
12 simplr 497 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B ) )  ->  B  e.  RR* )
13 pnfxr 7487 . . . . . . . 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 949 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B ) )  ->  C  <  B
)
16 pnfge 9194 . . . . . . . 8  |-  ( B  e.  RR*  ->  B  <_ +oo )
1716ad2antlr 473 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B ) )  ->  B  <_ +oo )
187, 12, 14, 15, 17xrltletrd 9211 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B ) )  ->  C  < +oo )
19 xrrebnd 9216 . . . . . . 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 888 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B ) )  ->  C  e.  RR )
2221, 10, 153jca 1121 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B ) )  ->  ( C  e.  RR  /\  A  <_  C  /\  C  <  B
) )
2322ex 113 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( ( C  e. 
RR*  /\  A  <_  C  /\  C  <  B
)  ->  ( C  e.  RR  /\  A  <_  C  /\  C  <  B
) ) )
24 rexr 7480 . . . 4  |-  ( C  e.  RR  ->  C  e.  RR* )
25243anim1i 1127 . . 3  |-  ( ( C  e.  RR  /\  A  <_  C  /\  C  <  B )  ->  ( C  e.  RR*  /\  A  <_  C  /\  C  < 
B ) )
2623, 25impbid1 140 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( ( C  e. 
RR*  /\  A  <_  C  /\  C  <  B
)  <->  ( C  e.  RR  /\  A  <_  C  /\  C  <  B
) ) )
273, 26bitrd 186 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 102    <-> wb 103    /\ w3a 922    e. wcel 1436   class class class wbr 3822  (class class class)co 5615   RRcr 7296   +oocpnf 7466   -oocmnf 7467   RR*cxr 7468    < clt 7469    <_ cle 7470   [,)cico 9243
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3934  ax-pow 3986  ax-pr 4012  ax-un 4236  ax-setind 4328  ax-cnex 7383  ax-resscn 7384  ax-pre-ltirr 7404  ax-pre-ltwlin 7405  ax-pre-lttrn 7406
This theorem depends on definitions:  df-bi 115  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-nel 2347  df-ral 2360  df-rex 2361  df-rab 2364  df-v 2617  df-sbc 2830  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3639  df-br 3823  df-opab 3877  df-id 4096  df-po 4099  df-iso 4100  df-xp 4419  df-rel 4420  df-cnv 4421  df-co 4422  df-dm 4423  df-iota 4948  df-fun 4985  df-fv 4991  df-ov 5618  df-oprab 5619  df-mpt2 5620  df-pnf 7471  df-mnf 7472  df-xr 7473  df-ltxr 7474  df-le 7475  df-ico 9247
This theorem is referenced by:  icossre  9307  elicopnf  9322  icoshft  9342  modqelico  9672  mulqaddmodid  9702  modqmuladdim  9705  addmodid  9710  icodiamlt  10512
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