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Theorem elioopnf 11621
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 11324 . . 3  |- +oo  e.  RR*
2 elioo2 11573 . . 3  |-  ( ( A  e.  RR*  /\ +oo  e.  RR* )  ->  ( B  e.  ( A (,) +oo )  <->  ( B  e.  RR  /\  A  < 
B  /\  B  < +oo ) ) )
31, 2mpan2 669 . 2  |-  ( A  e.  RR*  ->  ( B  e.  ( A (,) +oo )  <->  ( B  e.  RR  /\  A  < 
B  /\  B  < +oo ) ) )
4 df-3an 973 . . 3  |-  ( ( B  e.  RR  /\  A  <  B  /\  B  < +oo )  <->  ( ( B  e.  RR  /\  A  <  B )  /\  B  < +oo ) )
5 ltpnf 11334 . . . . 5  |-  ( B  e.  RR  ->  B  < +oo )
65adantr 463 . . . 4  |-  ( ( B  e.  RR  /\  A  <  B )  ->  B  < +oo )
76pm4.71i 630 . . 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 367    /\ w3a 971    e. wcel 1823   class class class wbr 4439  (class class class)co 6270   RRcr 9480   +oocpnf 9614   RR*cxr 9616    < clt 9617   (,)cioo 11532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-pre-lttri 9555  ax-pre-lttrn 9556
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-po 4789  df-so 4790  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-ioo 11536
This theorem is referenced by:  mbfmulc2lem  22220  mbfposr  22225  ismbf3d  22227  mbfaddlem  22233  mbfsup  22237  itg2gt0  22333  itg2cnlem1  22334  itg2cnlem2  22335  lhop2  22582  dvfsumlem2  22594  dvfsumlem3  22595  dvfsumrlimge0  22597  dvfsumrlim  22598  dvfsumrlim2  22599  pntpbnd1a  23968  pntpbnd2  23970  pntibndlem2  23974  pntibndlem3  23975  pntlemi  23987  pntlemo  23990  itg2addnclem2  30307  iblabsnclem  30318  ftc1anclem1  30330  ftc1anclem6  30335  rfcnpre1  31634  regt1loggt0  33411  rege1logbrege0  33433  rege1logbzge0  33434
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