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Theorem bsi 24667
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 10619 . 2  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
2 ffn 5246 . 2  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  (,)  Fn  ( RR*  X.  RR* )
)
3 ovelrn 5848 . 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 2510   ~Pcpw 3530    X. cxp 4578   ran crn 4581    Fn wfn 4587   -->wf 4588  (class class class)co 5710   RRcr 8616   RR*cxr 8746   (,)cioo 10534
This theorem is referenced by:  intvconlem1  24869
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 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-cnex 8673  ax-resscn 8674  ax-pre-lttri 8691  ax-pre-lttrn 8692
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-po 4207  df-so 4208  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-ioo 10538
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