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Theorem dfrp2 10264
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 9780 . . . . . 6  |-  ( x  e.  RR  ->  x  < +oo )
21adantr 276 . . . . 5  |-  ( ( x  e.  RR  /\  0  <  x )  ->  x  < +oo )
32pm4.71i 391 . . . 4  |-  ( ( x  e.  RR  /\  0  <  x )  <->  ( (
x  e.  RR  /\  0  <  x )  /\  x  < +oo ) )
4 df-3an 980 . . . 4  |-  ( ( x  e.  RR  /\  0  <  x  /\  x  < +oo )  <->  ( (
x  e.  RR  /\  0  <  x )  /\  x  < +oo ) )
53, 4bitr4i 187 . . 3  |-  ( ( x  e.  RR  /\  0  <  x )  <->  ( x  e.  RR  /\  0  < 
x  /\  x  < +oo ) )
6 elrp 9655 . . 3  |-  ( x  e.  RR+  <->  ( x  e.  RR  /\  0  < 
x ) )
7 0xr 8004 . . . 4  |-  0  e.  RR*
8 pnfxr 8010 . . . 4  |- +oo  e.  RR*
9 elioo2 9921 . . . 4  |-  ( ( 0  e.  RR*  /\ +oo  e.  RR* )  ->  (
x  e.  ( 0 (,) +oo )  <->  ( x  e.  RR  /\  0  < 
x  /\  x  < +oo ) ) )
107, 8, 9mp2an 426 . . 3  |-  ( x  e.  ( 0 (,) +oo )  <->  ( x  e.  RR  /\  0  < 
x  /\  x  < +oo ) )
115, 6, 103bitr4i 212 . 2  |-  ( x  e.  RR+  <->  x  e.  (
0 (,) +oo )
)
1211eqriv 2174 1  |-  RR+  =  ( 0 (,) +oo )
Colors of variables: wff set class
Syntax hints:    /\ wa 104    <-> wb 105    /\ w3a 978    = wceq 1353    e. wcel 2148   class class class wbr 4004  (class class class)co 5875   RRcr 7810   0cc0 7811   +oocpnf 7989   RR*cxr 7991    < clt 7992   RR+crp 9653   (,)cioo 9888
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-cnex 7902  ax-resscn 7903  ax-1re 7905  ax-addrcl 7908  ax-rnegex 7920  ax-pre-ltirr 7923  ax-pre-ltwlin 7924  ax-pre-lttrn 7925
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2740  df-sbc 2964  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-id 4294  df-po 4297  df-iso 4298  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-iota 5179  df-fun 5219  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-pnf 7994  df-mnf 7995  df-xr 7996  df-ltxr 7997  df-le 7998  df-rp 9654  df-ioo 9892
This theorem is referenced by: (None)
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