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Theorem elioopnf 10998
Description: Membership in an unbounded interval of extended reals. (Contributed by Mario Carneiro, 18-Jun-2014.)
Assertion
Ref Expression
elioopnf  |-  ( A  e.  RR*  ->  ( B  e.  ( A (,)  +oo )  <->  ( B  e.  RR  /\  A  < 
B ) ) )

Proof of Theorem elioopnf
StepHypRef Expression
1 pnfxr 10713 . . 3  |-  +oo  e.  RR*
2 elioo2 10957 . . 3  |-  ( ( A  e.  RR*  /\  +oo  e.  RR* )  ->  ( B  e.  ( A (,)  +oo )  <->  ( B  e.  RR  /\  A  < 
B  /\  B  <  +oo ) ) )
31, 2mpan2 653 . 2  |-  ( A  e.  RR*  ->  ( B  e.  ( A (,)  +oo )  <->  ( B  e.  RR  /\  A  < 
B  /\  B  <  +oo ) ) )
4 df-3an 938 . . 3  |-  ( ( B  e.  RR  /\  A  <  B  /\  B  <  +oo )  <->  ( ( B  e.  RR  /\  A  <  B )  /\  B  <  +oo ) )
5 ltpnf 10721 . . . . 5  |-  ( B  e.  RR  ->  B  <  +oo )
65adantr 452 . . . 4  |-  ( ( B  e.  RR  /\  A  <  B )  ->  B  <  +oo )
76pm4.71i 614 . . 3  |-  ( ( B  e.  RR  /\  A  <  B )  <->  ( ( B  e.  RR  /\  A  <  B )  /\  B  <  +oo ) )
84, 7bitr4i 244 . 2  |-  ( ( B  e.  RR  /\  A  <  B  /\  B  <  +oo )  <->  ( B  e.  RR  /\  A  < 
B ) )
93, 8syl6bb 253 1  |-  ( A  e.  RR*  ->  ( B  e.  ( A (,)  +oo )  <->  ( B  e.  RR  /\  A  < 
B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    e. wcel 1725   class class class wbr 4212  (class class class)co 6081   RRcr 8989    +oocpnf 9117   RR*cxr 9119    < clt 9120   (,)cioo 10916
This theorem is referenced by:  mbfmulc2lem  19539  mbfposr  19544  ismbf3d  19546  mbfaddlem  19552  mbfsup  19556  itg2gt0  19652  itg2cnlem1  19653  itg2cnlem2  19654  lhop2  19899  dvfsumlem2  19911  dvfsumlem3  19912  dvfsumrlimge0  19914  dvfsumrlim  19915  dvfsumrlim2  19916  pntpbnd1a  21279  pntpbnd2  21281  pntibndlem2  21285  pntibndlem3  21286  pntlemi  21298  pntlemo  21301  itg2addnclem2  26257  iblabsnclem  26268  ftc1anclem1  26280  ftc1anclem6  26285  rfcnpre1  27666
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-pre-lttri 9064  ax-pre-lttrn 9065
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-po 4503  df-so 4504  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-ioo 10920
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