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Theorem ico0 10262
Description: An empty open interval of extended reals. (Contributed by FL, 30-May-2014.)
Assertion
Ref Expression
ico0  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A [,) B
)  =  (/)  <->  B  <_  A ) )

Proof of Theorem ico0
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 icoval 9919 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A [,) B )  =  { x  e.  RR*  |  ( A  <_  x  /\  x  <  B ) } )
21eqeq1d 2186 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A [,) B
)  =  (/)  <->  { x  e.  RR*  |  ( A  <_  x  /\  x  <  B ) }  =  (/) ) )
3 xrlelttr 9806 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  x  e.  RR*  /\  B  e. 
RR* )  ->  (
( A  <_  x  /\  x  <  B )  ->  A  <  B
) )
433com23 1209 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  x  e. 
RR* )  ->  (
( A  <_  x  /\  x  <  B )  ->  A  <  B
) )
543expa 1203 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  x  e.  RR* )  ->  ( ( A  <_  x  /\  x  <  B
)  ->  A  <  B ) )
65rexlimdva 2594 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( E. x  e.  RR*  ( A  <_  x  /\  x  <  B )  ->  A  <  B ) )
7 qbtwnxr 10258 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) )
8 qre 9625 . . . . . . . . . . . 12  |-  ( x  e.  QQ  ->  x  e.  RR )
98rexrd 8007 . . . . . . . . . . 11  |-  ( x  e.  QQ  ->  x  e.  RR* )
109a1i 9 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR*  /\  ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B ) )  -> 
( x  e.  QQ  ->  x  e.  RR* )
)
11 simpr1 1003 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR*  /\  ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B ) )  ->  A  e.  RR* )
12 simpl 109 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR*  /\  ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B ) )  ->  x  e.  RR* )
13 xrltle 9798 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  RR*  /\  x  e.  RR* )  ->  ( A  <  x  ->  A  <_  x ) )
1411, 12, 13syl2anc 411 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR*  /\  ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B ) )  -> 
( A  <  x  ->  A  <_  x )
)
1514anim1d 336 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR*  /\  ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B ) )  -> 
( ( A  < 
x  /\  x  <  B )  ->  ( A  <_  x  /\  x  < 
B ) ) )
1610, 15anim12d 335 . . . . . . . . . . . 12  |-  ( ( x  e.  RR*  /\  ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B ) )  -> 
( ( x  e.  QQ  /\  ( A  <  x  /\  x  <  B ) )  -> 
( x  e.  RR*  /\  ( A  <_  x  /\  x  <  B ) ) ) )
1716ex 115 . . . . . . . . . . 11  |-  ( x  e.  RR*  ->  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  (
( x  e.  QQ  /\  ( A  <  x  /\  x  <  B ) )  ->  ( x  e.  RR*  /\  ( A  <_  x  /\  x  <  B ) ) ) ) )
189, 17syl 14 . . . . . . . . . 10  |-  ( x  e.  QQ  ->  (
( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  ->  (
( x  e.  QQ  /\  ( A  <  x  /\  x  <  B ) )  ->  ( x  e.  RR*  /\  ( A  <_  x  /\  x  <  B ) ) ) ) )
1918adantr 276 . . . . . . . . 9  |-  ( ( x  e.  QQ  /\  ( A  <  x  /\  x  <  B ) )  ->  ( ( A  e.  RR*  /\  B  e. 
RR*  /\  A  <  B )  ->  ( (
x  e.  QQ  /\  ( A  <  x  /\  x  <  B ) )  ->  ( x  e. 
RR*  /\  ( A  <_  x  /\  x  < 
B ) ) ) ) )
2019pm2.43b 52 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  (
( x  e.  QQ  /\  ( A  <  x  /\  x  <  B ) )  ->  ( x  e.  RR*  /\  ( A  <_  x  /\  x  <  B ) ) ) )
2120reximdv2 2576 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  ( E. x  e.  QQ  ( A  <  x  /\  x  <  B )  ->  E. x  e.  RR*  ( A  <_  x  /\  x  <  B ) ) )
227, 21mpd 13 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  E. x  e.  RR*  ( A  <_  x  /\  x  <  B
) )
23223expia 1205 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  ->  E. x  e.  RR*  ( A  <_  x  /\  x  <  B
) ) )
246, 23impbid 129 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( E. x  e.  RR*  ( A  <_  x  /\  x  <  B )  <->  A  <  B ) )
2524notbid 667 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( -.  E. x  e.  RR*  ( A  <_  x  /\  x  <  B )  <->  -.  A  <  B ) )
26 rabeq0 3453 . . . . 5  |-  ( { x  e.  RR*  |  ( A  <_  x  /\  x  <  B ) }  =  (/)  <->  A. x  e.  RR*  -.  ( A  <_  x  /\  x  <  B ) )
27 ralnex 2465 . . . . 5  |-  ( A. x  e.  RR*  -.  ( A  <_  x  /\  x  <  B )  <->  -.  E. x  e.  RR*  ( A  <_  x  /\  x  <  B
) )
2826, 27bitri 184 . . . 4  |-  ( { x  e.  RR*  |  ( A  <_  x  /\  x  <  B ) }  =  (/)  <->  -.  E. x  e.  RR*  ( A  <_  x  /\  x  <  B
) )
2928a1i 9 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( { x  e.  RR*  |  ( A  <_  x  /\  x  <  B ) }  =  (/)  <->  -.  E. x  e.  RR*  ( A  <_  x  /\  x  <  B
) ) )
30 xrlenlt 8022 . . . 4  |-  ( ( B  e.  RR*  /\  A  e.  RR* )  ->  ( B  <_  A  <->  -.  A  <  B ) )
3130ancoms 268 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( B  <_  A  <->  -.  A  <  B ) )
3225, 29, 313bitr4d 220 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( { x  e.  RR*  |  ( A  <_  x  /\  x  <  B ) }  =  (/)  <->  B  <_  A ) )
332, 32bitrd 188 1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A [,) B
)  =  (/)  <->  B  <_  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 978    = wceq 1353    e. wcel 2148   A.wral 2455   E.wrex 2456   {crab 2459   (/)c0 3423   class class class wbr 4004  (class class class)co 5875   RR*cxr 7991    < clt 7992    <_ cle 7993   QQcq 9619   [,)cico 9890
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-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-mulrcl 7910  ax-addcom 7911  ax-mulcom 7912  ax-addass 7913  ax-mulass 7914  ax-distr 7915  ax-i2m1 7916  ax-0lt1 7917  ax-1rid 7918  ax-0id 7919  ax-rnegex 7920  ax-precex 7921  ax-cnre 7922  ax-pre-ltirr 7923  ax-pre-ltwlin 7924  ax-pre-lttrn 7925  ax-pre-apti 7926  ax-pre-ltadd 7927  ax-pre-mulgt0 7928  ax-pre-mulext 7929  ax-arch 7930
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-reu 2462  df-rmo 2463  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  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-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-pnf 7994  df-mnf 7995  df-xr 7996  df-ltxr 7997  df-le 7998  df-sub 8130  df-neg 8131  df-reap 8532  df-ap 8539  df-div 8630  df-inn 8920  df-2 8978  df-n0 9177  df-z 9254  df-uz 9529  df-q 9620  df-rp 9654  df-ico 9894
This theorem is referenced by: (None)
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