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Theorem bsi 24902
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 10737 . 2  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
2 ffn 5356 . 2  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  (,)  Fn  ( RR*  X.  RR* )
)
3 ovelrn 5959 . 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 1625    e. wcel 1687   E.wrex 2547   ~Pcpw 3628    X. cxp 4688   ran crn 4691    Fn wfn 5218   -->wf 5219  (class class class)co 5821   RRcr 8733   RR*cxr 8863   (,)cioo 10652
This theorem is referenced by:  intvconlem1  25104
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1638  ax-8 1646  ax-13 1689  ax-14 1691  ax-6 1706  ax-7 1711  ax-11 1718  ax-12 1870  ax-ext 2267  ax-sep 4144  ax-nul 4152  ax-pow 4189  ax-pr 4215  ax-un 4513  ax-cnex 8790  ax-resscn 8791  ax-pre-lttri 8808  ax-pre-lttrn 8809
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1531  df-nf 1534  df-sb 1633  df-eu 2150  df-mo 2151  df-clab 2273  df-cleq 2279  df-clel 2282  df-nfc 2411  df-ne 2451  df-nel 2452  df-ral 2551  df-rex 2552  df-rab 2555  df-v 2793  df-sbc 2995  df-csb 3085  df-dif 3158  df-un 3160  df-in 3162  df-ss 3169  df-nul 3459  df-if 3569  df-pw 3630  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3831  df-iun 3910  df-br 4027  df-opab 4081  df-mpt 4082  df-id 4310  df-po 4315  df-so 4316  df-xp 4696  df-rel 4697  df-cnv 4698  df-co 4699  df-dm 4700  df-rn 4701  df-res 4702  df-ima 4703  df-fun 5225  df-fn 5226  df-f 5227  df-f1 5228  df-fo 5229  df-f1o 5230  df-fv 5231  df-ov 5824  df-oprab 5825  df-mpt2 5826  df-1st 6085  df-2nd 6086  df-er 6657  df-en 6861  df-dom 6862  df-sdom 6863  df-pnf 8866  df-mnf 8867  df-xr 8868  df-ltxr 8869  df-le 8870  df-ioo 10656
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