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Theorem iocinico 30812
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 3483 . . . . . 6  |-  ( ( A (,] B )  i^i  ( B [,) C ) )  =  { x  |  ( x  e.  ( A (,] B )  /\  x  e.  ( B [,) C ) ) }
2 elioc1 11571 . . . . . . . . . . . 12  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
x  e.  ( A (,] B )  <->  ( x  e.  RR*  /\  A  < 
x  /\  x  <_  B ) ) )
323adant3 1016 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
x  e.  ( A (,] B )  <->  ( x  e.  RR*  /\  A  < 
x  /\  x  <_  B ) ) )
4 3simpb 994 . . . . . . . . . . 11  |-  ( ( x  e.  RR*  /\  A  <  x  /\  x  <_  B )  ->  (
x  e.  RR*  /\  x  <_  B ) )
53, 4syl6bi 228 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
x  e.  ( A (,] B )  -> 
( x  e.  RR*  /\  x  <_  B )
) )
6 elico1 11572 . . . . . . . . . . . 12  |-  ( ( B  e.  RR*  /\  C  e.  RR* )  ->  (
x  e.  ( B [,) C )  <->  ( x  e.  RR*  /\  B  <_  x  /\  x  <  C
) ) )
763adant1 1014 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
x  e.  ( B [,) C )  <->  ( x  e.  RR*  /\  B  <_  x  /\  x  <  C
) ) )
8 3simpa 993 . . . . . . . . . . 11  |-  ( ( x  e.  RR*  /\  B  <_  x  /\  x  < 
C )  ->  (
x  e.  RR*  /\  B  <_  x ) )
97, 8syl6bi 228 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
x  e.  ( B [,) C )  -> 
( x  e.  RR*  /\  B  <_  x )
) )
105, 9anim12d 563 . . . . . . . . 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 753 . . . . . . . . . 10  |-  ( ( ( x  e.  RR*  /\  x  <_  B )  /\  ( x  e.  RR*  /\  B  <_  x )
)  ->  x  e.  RR* )
12 simprr 756 . . . . . . . . . 10  |-  ( ( ( x  e.  RR*  /\  x  <_  B )  /\  ( x  e.  RR*  /\  B  <_  x )
)  ->  B  <_  x )
13 simplr 754 . . . . . . . . . 10  |-  ( ( ( x  e.  RR*  /\  x  <_  B )  /\  ( x  e.  RR*  /\  B  <_  x )
)  ->  x  <_  B )
1411, 12, 133jca 1176 . . . . . . . . 9  |-  ( ( ( x  e.  RR*  /\  x  <_  B )  /\  ( x  e.  RR*  /\  B  <_  x )
)  ->  ( x  e.  RR*  /\  B  <_  x  /\  x  <_  B
) )
1510, 14syl6 33 . . . . . . . 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 11573 . . . . . . . . . 10  |-  ( ( B  e.  RR*  /\  B  e.  RR* )  ->  (
x  e.  ( B [,] B )  <->  ( x  e.  RR*  /\  B  <_  x  /\  x  <_  B
) ) )
1716anidms 645 . . . . . . . . 9  |-  ( B  e.  RR*  ->  ( x  e.  ( B [,] B )  <->  ( x  e.  RR*  /\  B  <_  x  /\  x  <_  B
) ) )
18173ad2ant2 1018 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
x  e.  ( B [,] B )  <->  ( x  e.  RR*  /\  B  <_  x  /\  x  <_  B
) ) )
1915, 18sylibrd 234 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( x  e.  ( A (,] B )  /\  x  e.  ( B [,) C ) )  ->  x  e.  ( B [,] B ) ) )
2019ss2abdv 3573 . . . . . 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 3548 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( A (,] B
)  i^i  ( B [,) C ) )  C_  { x  |  x  e.  ( B [,] B
) } )
22 abid2 2607 . . . . 5  |-  { x  |  x  e.  ( B [,] B ) }  =  ( B [,] B )
2321, 22syl6sseq 3550 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( A (,] B
)  i^i  ( B [,) C ) )  C_  ( B [,] B ) )
2423adantr 465 . . 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 11574 . . . . 5  |-  ( B  e.  RR*  ->  ( B [,] B )  =  { B } )
26253ad2ant2 1018 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( B [,] B )  =  { B } )
2726adantr 465 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  -> 
( B [,] B
)  =  { B } )
2824, 27sseqtrd 3540 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  -> 
( ( A (,] B )  i^i  ( B [,) C ) ) 
C_  { B }
)
29 simpl2 1000 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  ->  B  e.  RR* )
30 simprl 755 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  ->  A  <  B )
31 xrleid 11356 . . . . . 6  |-  ( B  e.  RR*  ->  B  <_  B )
3229, 31syl 16 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  ->  B  <_  B )
33 elioc1 11571 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( B  e.  ( A (,] B )  <->  ( B  e.  RR*  /\  A  < 
B  /\  B  <_  B ) ) )
34333adant3 1016 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( B  e.  ( A (,] B )  <->  ( B  e.  RR*  /\  A  < 
B  /\  B  <_  B ) ) )
3534adantr 465 . . . . 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 1179 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  ->  B  e.  ( A (,] B ) )
37 simprr 756 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  ->  B  <  C )
38 elico1 11572 . . . . . . 7  |-  ( ( B  e.  RR*  /\  C  e.  RR* )  ->  ( B  e.  ( B [,) C )  <->  ( B  e.  RR*  /\  B  <_  B  /\  B  <  C
) ) )
39383adant1 1014 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( B  e.  ( B [,) C )  <->  ( B  e.  RR*  /\  B  <_  B  /\  B  <  C
) ) )
4039adantr 465 . . . . 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 1179 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  ->  B  e.  ( B [,) C ) )
4236, 41elind 3688 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  ->  B  e.  ( ( A (,] B )  i^i  ( B [,) C
) ) )
4342snssd 4172 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  ->  { B }  C_  (
( A (,] B
)  i^i  ( B [,) C ) ) )
4428, 43eqssd 3521 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 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   {cab 2452    i^i cin 3475    C_ wss 3476   {csn 4027   class class class wbr 4447  (class class class)co 6284   RR*cxr 9627    < clt 9628    <_ cle 9629   (,]cioc 11530   [,)cico 11531   [,]cicc 11532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-pre-lttri 9566  ax-pre-lttrn 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-ioc 11534  df-ico 11535  df-icc 11536
This theorem is referenced by: (None)
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