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Theorem elioopnf 11000
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 10715 . . 3  |-  +oo  e.  RR*
2 elioo2 10959 . . 3  |-  ( ( A  e.  RR*  /\  +oo  e.  RR* )  ->  ( B  e.  ( A (,)  +oo )  <->  ( B  e.  RR  /\  A  < 
B  /\  B  <  +oo ) ) )
31, 2mpan2 654 . 2  |-  ( A  e.  RR*  ->  ( B  e.  ( A (,)  +oo )  <->  ( B  e.  RR  /\  A  < 
B  /\  B  <  +oo ) ) )
4 df-3an 939 . . 3  |-  ( ( B  e.  RR  /\  A  <  B  /\  B  <  +oo )  <->  ( ( B  e.  RR  /\  A  <  B )  /\  B  <  +oo ) )
5 ltpnf 10723 . . . . 5  |-  ( B  e.  RR  ->  B  <  +oo )
65adantr 453 . . . 4  |-  ( ( B  e.  RR  /\  A  <  B )  ->  B  <  +oo )
76pm4.71i 615 . . 3  |-  ( ( B  e.  RR  /\  A  <  B )  <->  ( ( B  e.  RR  /\  A  <  B )  /\  B  <  +oo ) )
84, 7bitr4i 245 . 2  |-  ( ( B  e.  RR  /\  A  <  B  /\  B  <  +oo )  <->  ( B  e.  RR  /\  A  < 
B ) )
93, 8syl6bb 254 1  |-  ( A  e.  RR*  ->  ( B  e.  ( A (,)  +oo )  <->  ( B  e.  RR  /\  A  < 
B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    e. wcel 1726   class class class wbr 4214  (class class class)co 6083   RRcr 8991    +oocpnf 9119   RR*cxr 9121    < clt 9122   (,)cioo 10918
This theorem is referenced by:  mbfmulc2lem  19541  mbfposr  19546  ismbf3d  19548  mbfaddlem  19554  mbfsup  19558  itg2gt0  19654  itg2cnlem1  19655  itg2cnlem2  19656  lhop2  19901  dvfsumlem2  19913  dvfsumlem3  19914  dvfsumrlimge0  19916  dvfsumrlim  19917  dvfsumrlim2  19918  pntpbnd1a  21281  pntpbnd2  21283  pntibndlem2  21287  pntibndlem3  21288  pntlemi  21300  pntlemo  21303  itg2addnclem2  26259  iblabsnclem  26270  ftc1anclem1  26282  ftc1anclem6  26287  rfcnpre1  27668
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-pre-lttri 9066  ax-pre-lttrn 9067
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-po 4505  df-so 4506  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-ioo 10922
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