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Theorem elioomnf 9986
Description: Membership in an unbounded interval of extended reals. (Contributed by Mario Carneiro, 18-Jun-2014.)
Assertion
Ref Expression
elioomnf  |-  ( A  e.  RR*  ->  ( B  e.  ( -oo (,) A )  <->  ( B  e.  RR  /\  B  < 
A ) ) )

Proof of Theorem elioomnf
StepHypRef Expression
1 mnfxr 8032 . . 3  |- -oo  e.  RR*
2 elioo2 9939 . . 3  |-  ( ( -oo  e.  RR*  /\  A  e.  RR* )  ->  ( B  e.  ( -oo (,) A )  <->  ( B  e.  RR  /\ -oo  <  B  /\  B  <  A
) ) )
31, 2mpan 424 . 2  |-  ( A  e.  RR*  ->  ( B  e.  ( -oo (,) A )  <->  ( B  e.  RR  /\ -oo  <  B  /\  B  <  A
) ) )
4 an32 562 . . 3  |-  ( ( ( B  e.  RR  /\ -oo  <  B )  /\  B  <  A )  <->  ( ( B  e.  RR  /\  B  <  A )  /\ -oo  <  B ) )
5 df-3an 982 . . 3  |-  ( ( B  e.  RR  /\ -oo 
<  B  /\  B  < 
A )  <->  ( ( B  e.  RR  /\ -oo  <  B )  /\  B  <  A ) )
6 mnflt 9801 . . . . 5  |-  ( B  e.  RR  -> -oo  <  B )
76adantr 276 . . . 4  |-  ( ( B  e.  RR  /\  B  <  A )  -> -oo  <  B )
87pm4.71i 391 . . 3  |-  ( ( B  e.  RR  /\  B  <  A )  <->  ( ( B  e.  RR  /\  B  <  A )  /\ -oo  <  B ) )
94, 5, 83bitr4i 212 . 2  |-  ( ( B  e.  RR  /\ -oo 
<  B  /\  B  < 
A )  <->  ( B  e.  RR  /\  B  < 
A ) )
103, 9bitrdi 196 1  |-  ( A  e.  RR*  ->  ( B  e.  ( -oo (,) A )  <->  ( B  e.  RR  /\  B  < 
A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 980    e. wcel 2160   class class class wbr 4018  (class class class)co 5891   RRcr 7828   -oocmnf 8008   RR*cxr 8009    < clt 8010   (,)cioo 9906
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-cnex 7920  ax-resscn 7921  ax-pre-ltirr 7941  ax-pre-ltwlin 7942  ax-pre-lttrn 7943
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4308  df-po 4311  df-iso 4312  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-iota 5193  df-fun 5233  df-fv 5239  df-ov 5894  df-oprab 5895  df-mpo 5896  df-pnf 8012  df-mnf 8013  df-xr 8014  df-ltxr 8015  df-le 8016  df-ioo 9910
This theorem is referenced by:  reopnap  14435
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