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Theorem elioopnf 9143
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 7310 . . 3  |- +oo  e.  RR*
2 elioo2 9097 . . 3  |-  ( ( A  e.  RR*  /\ +oo  e.  RR* )  ->  ( B  e.  ( A (,) +oo )  <->  ( B  e.  RR  /\  A  < 
B  /\  B  < +oo ) ) )
31, 2mpan2 416 . 2  |-  ( A  e.  RR*  ->  ( B  e.  ( A (,) +oo )  <->  ( B  e.  RR  /\  A  < 
B  /\  B  < +oo ) ) )
4 df-3an 922 . . 3  |-  ( ( B  e.  RR  /\  A  <  B  /\  B  < +oo )  <->  ( ( B  e.  RR  /\  A  <  B )  /\  B  < +oo ) )
5 ltpnf 9009 . . . . 5  |-  ( B  e.  RR  ->  B  < +oo )
65adantr 270 . . . 4  |-  ( ( B  e.  RR  /\  A  <  B )  ->  B  < +oo )
76pm4.71i 383 . . 3  |-  ( ( B  e.  RR  /\  A  <  B )  <->  ( ( B  e.  RR  /\  A  <  B )  /\  B  < +oo ) )
84, 7bitr4i 185 . 2  |-  ( ( B  e.  RR  /\  A  <  B  /\  B  < +oo )  <->  ( B  e.  RR  /\  A  < 
B ) )
93, 8syl6bb 194 1  |-  ( A  e.  RR*  ->  ( B  e.  ( A (,) +oo )  <->  ( B  e.  RR  /\  A  < 
B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    /\ w3a 920    e. wcel 1434   class class class wbr 3806  (class class class)co 5565   RRcr 7119   +oocpnf 7289   RR*cxr 7291    < clt 7292   (,)cioo 9064
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3917  ax-pow 3969  ax-pr 3993  ax-un 4217  ax-setind 4309  ax-cnex 7206  ax-resscn 7207  ax-pre-ltirr 7227  ax-pre-ltwlin 7228  ax-pre-lttrn 7229
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2613  df-sbc 2826  df-dif 2985  df-un 2987  df-in 2989  df-ss 2996  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-br 3807  df-opab 3861  df-id 4077  df-po 4080  df-iso 4081  df-xp 4398  df-rel 4399  df-cnv 4400  df-co 4401  df-dm 4402  df-iota 4918  df-fun 4955  df-fv 4961  df-ov 5568  df-oprab 5569  df-mpt2 5570  df-pnf 7294  df-mnf 7295  df-xr 7296  df-ltxr 7297  df-le 7298  df-ioo 9068
This theorem is referenced by: (None)
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