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Theorem iocinico 35795
Description: The intersection of two sets that meet at a point is that point. (Contributed by Jon Pennant, 12-Jun-2019.)
Assertion
Ref Expression
iocinico  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  -> 
( ( A (,] B )  i^i  ( B [,) C ) )  =  { B }
)

Proof of Theorem iocinico
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 df-in 3449 . . . . . 6  |-  ( ( A (,] B )  i^i  ( B [,) C ) )  =  { x  |  ( x  e.  ( A (,] B )  /\  x  e.  ( B [,) C ) ) }
2 elioc1 11678 . . . . . . . . . . . 12  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
x  e.  ( A (,] B )  <->  ( x  e.  RR*  /\  A  < 
x  /\  x  <_  B ) ) )
323adant3 1025 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
x  e.  ( A (,] B )  <->  ( x  e.  RR*  /\  A  < 
x  /\  x  <_  B ) ) )
4 3simpb 1003 . . . . . . . . . . 11  |-  ( ( x  e.  RR*  /\  A  <  x  /\  x  <_  B )  ->  (
x  e.  RR*  /\  x  <_  B ) )
53, 4syl6bi 231 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
x  e.  ( A (,] B )  -> 
( x  e.  RR*  /\  x  <_  B )
) )
6 elico1 11679 . . . . . . . . . . . 12  |-  ( ( B  e.  RR*  /\  C  e.  RR* )  ->  (
x  e.  ( B [,) C )  <->  ( x  e.  RR*  /\  B  <_  x  /\  x  <  C
) ) )
763adant1 1023 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
x  e.  ( B [,) C )  <->  ( x  e.  RR*  /\  B  <_  x  /\  x  <  C
) ) )
8 3simpa 1002 . . . . . . . . . . 11  |-  ( ( x  e.  RR*  /\  B  <_  x  /\  x  < 
C )  ->  (
x  e.  RR*  /\  B  <_  x ) )
97, 8syl6bi 231 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
x  e.  ( B [,) C )  -> 
( x  e.  RR*  /\  B  <_  x )
) )
105, 9anim12d 565 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( x  e.  ( A (,] B )  /\  x  e.  ( B [,) C ) )  ->  ( (
x  e.  RR*  /\  x  <_  B )  /\  (
x  e.  RR*  /\  B  <_  x ) ) ) )
11 simpll 758 . . . . . . . . . 10  |-  ( ( ( x  e.  RR*  /\  x  <_  B )  /\  ( x  e.  RR*  /\  B  <_  x )
)  ->  x  e.  RR* )
12 simprr 764 . . . . . . . . . 10  |-  ( ( ( x  e.  RR*  /\  x  <_  B )  /\  ( x  e.  RR*  /\  B  <_  x )
)  ->  B  <_  x )
13 simplr 760 . . . . . . . . . 10  |-  ( ( ( x  e.  RR*  /\  x  <_  B )  /\  ( x  e.  RR*  /\  B  <_  x )
)  ->  x  <_  B )
1411, 12, 133jca 1185 . . . . . . . . 9  |-  ( ( ( x  e.  RR*  /\  x  <_  B )  /\  ( x  e.  RR*  /\  B  <_  x )
)  ->  ( x  e.  RR*  /\  B  <_  x  /\  x  <_  B
) )
1510, 14syl6 34 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( x  e.  ( A (,] B )  /\  x  e.  ( B [,) C ) )  ->  ( x  e.  RR*  /\  B  <_  x  /\  x  <_  B
) ) )
16 elicc1 11680 . . . . . . . . . 10  |-  ( ( B  e.  RR*  /\  B  e.  RR* )  ->  (
x  e.  ( B [,] B )  <->  ( x  e.  RR*  /\  B  <_  x  /\  x  <_  B
) ) )
1716anidms 649 . . . . . . . . 9  |-  ( B  e.  RR*  ->  ( x  e.  ( B [,] B )  <->  ( x  e.  RR*  /\  B  <_  x  /\  x  <_  B
) ) )
18173ad2ant2 1027 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
x  e.  ( B [,] B )  <->  ( x  e.  RR*  /\  B  <_  x  /\  x  <_  B
) ) )
1915, 18sylibrd 237 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( x  e.  ( A (,] B )  /\  x  e.  ( B [,) C ) )  ->  x  e.  ( B [,] B ) ) )
2019ss2abdv 3540 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  { x  |  ( x  e.  ( A (,] B
)  /\  x  e.  ( B [,) C ) ) }  C_  { x  |  x  e.  ( B [,] B ) } )
211, 20syl5eqss 3514 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( A (,] B
)  i^i  ( B [,) C ) )  C_  { x  |  x  e.  ( B [,] B
) } )
22 abid2 2569 . . . . 5  |-  { x  |  x  e.  ( B [,] B ) }  =  ( B [,] B )
2321, 22syl6sseq 3516 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( A (,] B
)  i^i  ( B [,) C ) )  C_  ( B [,] B ) )
2423adantr 466 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  -> 
( ( A (,] B )  i^i  ( B [,) C ) ) 
C_  ( B [,] B ) )
25 iccid 11681 . . . . 5  |-  ( B  e.  RR*  ->  ( B [,] B )  =  { B } )
26253ad2ant2 1027 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( B [,] B )  =  { B } )
2726adantr 466 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  -> 
( B [,] B
)  =  { B } )
2824, 27sseqtrd 3506 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  -> 
( ( A (,] B )  i^i  ( B [,) C ) ) 
C_  { B }
)
29 simpl2 1009 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  ->  B  e.  RR* )
30 simprl 762 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  ->  A  <  B )
31 xrleid 11449 . . . . . 6  |-  ( B  e.  RR*  ->  B  <_  B )
3229, 31syl 17 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  ->  B  <_  B )
33 elioc1 11678 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( B  e.  ( A (,] B )  <->  ( B  e.  RR*  /\  A  < 
B  /\  B  <_  B ) ) )
34333adant3 1025 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( B  e.  ( A (,] B )  <->  ( B  e.  RR*  /\  A  < 
B  /\  B  <_  B ) ) )
3534adantr 466 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  -> 
( B  e.  ( A (,] B )  <-> 
( B  e.  RR*  /\  A  <  B  /\  B  <_  B ) ) )
3629, 30, 32, 35mpbir3and 1188 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  ->  B  e.  ( A (,] B ) )
37 simprr 764 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  ->  B  <  C )
38 elico1 11679 . . . . . . 7  |-  ( ( B  e.  RR*  /\  C  e.  RR* )  ->  ( B  e.  ( B [,) C )  <->  ( B  e.  RR*  /\  B  <_  B  /\  B  <  C
) ) )
39383adant1 1023 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( B  e.  ( B [,) C )  <->  ( B  e.  RR*  /\  B  <_  B  /\  B  <  C
) ) )
4039adantr 466 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  -> 
( B  e.  ( B [,) C )  <-> 
( B  e.  RR*  /\  B  <_  B  /\  B  <  C ) ) )
4129, 32, 37, 40mpbir3and 1188 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  ->  B  e.  ( B [,) C ) )
4236, 41elind 3656 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  ->  B  e.  ( ( A (,] B )  i^i  ( B [,) C
) ) )
4342snssd 4148 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  ->  { B }  C_  (
( A (,] B
)  i^i  ( B [,) C ) ) )
4428, 43eqssd 3487 1  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  -> 
( ( A (,] B )  i^i  ( B [,) C ) )  =  { B }
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870   {cab 2414    i^i cin 3441    C_ wss 3442   {csn 4002   class class class wbr 4426  (class class class)co 6305   RR*cxr 9673    < clt 9674    <_ cle 9675   (,]cioc 11636   [,)cico 11637   [,]cicc 11638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-pre-lttri 9612  ax-pre-lttrn 9613
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-po 4775  df-so 4776  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-ioc 11640  df-ico 11641  df-icc 11642
This theorem is referenced by: (None)
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