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Theorem bsi 24833
Description: Membership to the set of open intervals implied the existence of two bounds in the set of the extended reals. (Contributed by FL, 31-Oct-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)
Assertion
Ref Expression
bsi  |-  ( A  e.  ran  (,)  <->  E. x  e.  RR*  E. y  e. 
RR*  A  =  (
x (,) y ) )
Distinct variable group:    x, A, y

Proof of Theorem bsi
StepHypRef Expression
1 ioof 10672 . 2  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
2 ffn 5292 . 2  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  (,)  Fn  ( RR*  X.  RR* )
)
3 ovelrn 5895 . 2  |-  ( (,) 
Fn  ( RR*  X.  RR* )  ->  ( A  e. 
ran  (,)  <->  E. x  e.  RR*  E. y  e.  RR*  A  =  ( x (,) y ) ) )
41, 2, 3mp2b 11 1  |-  ( A  e.  ran  (,)  <->  E. x  e.  RR*  E. y  e. 
RR*  A  =  (
x (,) y ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    = wceq 1619    e. wcel 1621   E.wrex 2517   ~Pcpw 3566    X. cxp 4624   ran crn 4627    Fn wfn 4633   -->wf 4634  (class class class)co 5757   RRcr 8669   RR*cxr 8799   (,)cioo 10587
This theorem is referenced by:  intvconlem1  25035
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-cnex 8726  ax-resscn 8727  ax-pre-lttri 8744  ax-pre-lttrn 8745
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-po 4251  df-so 4252  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-1st 6021  df-2nd 6022  df-er 6593  df-en 6797  df-dom 6798  df-sdom 6799  df-pnf 8802  df-mnf 8803  df-xr 8804  df-ltxr 8805  df-le 8806  df-ioo 10591
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