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Theorem elioopnf 11499
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 11202 . . 3  |- +oo  e.  RR*
2 elioo2 11451 . . 3  |-  ( ( A  e.  RR*  /\ +oo  e.  RR* )  ->  ( B  e.  ( A (,) +oo )  <->  ( B  e.  RR  /\  A  < 
B  /\  B  < +oo ) ) )
31, 2mpan2 671 . 2  |-  ( A  e.  RR*  ->  ( B  e.  ( A (,) +oo )  <->  ( B  e.  RR  /\  A  < 
B  /\  B  < +oo ) ) )
4 df-3an 967 . . 3  |-  ( ( B  e.  RR  /\  A  <  B  /\  B  < +oo )  <->  ( ( B  e.  RR  /\  A  <  B )  /\  B  < +oo ) )
5 ltpnf 11212 . . . . 5  |-  ( B  e.  RR  ->  B  < +oo )
65adantr 465 . . . 4  |-  ( ( B  e.  RR  /\  A  <  B )  ->  B  < +oo )
76pm4.71i 632 . . 3  |-  ( ( B  e.  RR  /\  A  <  B )  <->  ( ( B  e.  RR  /\  A  <  B )  /\  B  < +oo ) )
84, 7bitr4i 252 . 2  |-  ( ( B  e.  RR  /\  A  <  B  /\  B  < +oo )  <->  ( B  e.  RR  /\  A  < 
B ) )
93, 8syl6bb 261 1  |-  ( A  e.  RR*  ->  ( B  e.  ( A (,) +oo )  <->  ( B  e.  RR  /\  A  < 
B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    e. wcel 1758   class class class wbr 4399  (class class class)co 6199   RRcr 9391   +oocpnf 9525   RR*cxr 9527    < clt 9528   (,)cioo 11410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-cnex 9448  ax-resscn 9449  ax-pre-lttri 9466  ax-pre-lttrn 9467
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-id 4743  df-po 4748  df-so 4749  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-1st 6686  df-2nd 6687  df-er 7210  df-en 7420  df-dom 7421  df-sdom 7422  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-ioo 11414
This theorem is referenced by:  mbfmulc2lem  21257  mbfposr  21262  ismbf3d  21264  mbfaddlem  21270  mbfsup  21274  itg2gt0  21370  itg2cnlem1  21371  itg2cnlem2  21372  lhop2  21619  dvfsumlem2  21631  dvfsumlem3  21632  dvfsumrlimge0  21634  dvfsumrlim  21635  dvfsumrlim2  21636  pntpbnd1a  22966  pntpbnd2  22968  pntibndlem2  22972  pntibndlem3  22973  pntlemi  22985  pntlemo  22988  itg2addnclem2  28591  iblabsnclem  28602  ftc1anclem1  28614  ftc1anclem6  28619  rfcnpre1  29888
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