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Theorem dfrp2 10158
Description: Alternate definition of the positive real numbers. (Contributed by Thierry Arnoux, 4-May-2020.)
Assertion
Ref Expression
dfrp2  |-  RR+  =  ( 0 (,) +oo )

Proof of Theorem dfrp2
StepHypRef Expression
1 ltpnf 9682 . . . . . 6  |-  ( x  e.  RR  ->  x  < +oo )
21adantr 274 . . . . 5  |-  ( ( x  e.  RR  /\  0  <  x )  ->  x  < +oo )
32pm4.71i 389 . . . 4  |-  ( ( x  e.  RR  /\  0  <  x )  <->  ( (
x  e.  RR  /\  0  <  x )  /\  x  < +oo ) )
4 df-3an 965 . . . 4  |-  ( ( x  e.  RR  /\  0  <  x  /\  x  < +oo )  <->  ( (
x  e.  RR  /\  0  <  x )  /\  x  < +oo ) )
53, 4bitr4i 186 . . 3  |-  ( ( x  e.  RR  /\  0  <  x )  <->  ( x  e.  RR  /\  0  < 
x  /\  x  < +oo ) )
6 elrp 9557 . . 3  |-  ( x  e.  RR+  <->  ( x  e.  RR  /\  0  < 
x ) )
7 0xr 7919 . . . 4  |-  0  e.  RR*
8 pnfxr 7925 . . . 4  |- +oo  e.  RR*
9 elioo2 9820 . . . 4  |-  ( ( 0  e.  RR*  /\ +oo  e.  RR* )  ->  (
x  e.  ( 0 (,) +oo )  <->  ( x  e.  RR  /\  0  < 
x  /\  x  < +oo ) ) )
107, 8, 9mp2an 423 . . 3  |-  ( x  e.  ( 0 (,) +oo )  <->  ( x  e.  RR  /\  0  < 
x  /\  x  < +oo ) )
115, 6, 103bitr4i 211 . 2  |-  ( x  e.  RR+  <->  x  e.  (
0 (,) +oo )
)
1211eqriv 2154 1  |-  RR+  =  ( 0 (,) +oo )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    /\ w3a 963    = wceq 1335    e. wcel 2128   class class class wbr 3965  (class class class)co 5821   RRcr 7726   0cc0 7727   +oocpnf 7904   RR*cxr 7906    < clt 7907   RR+crp 9555   (,)cioo 9787
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4495  ax-cnex 7818  ax-resscn 7819  ax-1re 7821  ax-addrcl 7824  ax-rnegex 7836  ax-pre-ltirr 7839  ax-pre-ltwlin 7840  ax-pre-lttrn 7841
This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-id 4253  df-po 4256  df-iso 4257  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-iota 5134  df-fun 5171  df-fv 5177  df-ov 5824  df-oprab 5825  df-mpo 5826  df-pnf 7909  df-mnf 7910  df-xr 7911  df-ltxr 7912  df-le 7913  df-rp 9556  df-ioo 9791
This theorem is referenced by: (None)
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