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Theorem dfrp2 10220
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 9737 . . . . . 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 975 . . . 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 9612 . . 3  |-  ( x  e.  RR+  <->  ( x  e.  RR  /\  0  < 
x ) )
7 0xr 7966 . . . 4  |-  0  e.  RR*
8 pnfxr 7972 . . . 4  |- +oo  e.  RR*
9 elioo2 9878 . . . 4  |-  ( ( 0  e.  RR*  /\ +oo  e.  RR* )  ->  (
x  e.  ( 0 (,) +oo )  <->  ( x  e.  RR  /\  0  < 
x  /\  x  < +oo ) ) )
107, 8, 9mp2an 424 . . 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 2167 1  |-  RR+  =  ( 0 (,) +oo )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    /\ w3a 973    = wceq 1348    e. wcel 2141   class class class wbr 3989  (class class class)co 5853   RRcr 7773   0cc0 7774   +oocpnf 7951   RR*cxr 7953    < clt 7954   RR+crp 9610   (,)cioo 9845
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1re 7868  ax-addrcl 7871  ax-rnegex 7883  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888
This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-po 4281  df-iso 4282  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-rp 9611  df-ioo 9849
This theorem is referenced by: (None)
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