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Theorem icoiccdif 37721
Description: Left closed, right open interval gotten by a closed iterval taking away the upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Assertion
Ref Expression
icoiccdif  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A [,) B )  =  ( ( A [,] B )  \  { B } ) )

Proof of Theorem icoiccdif
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 icossicc 11746 . . . . . . 7  |-  ( A [,) B )  C_  ( A [,] B )
21a1i 11 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A [,) B )  C_  ( A [,] B ) )
32sselda 3418 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  x  e.  ( A [,) B ) )  ->  x  e.  ( A [,] B ) )
4 elico1 11704 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
x  e.  ( A [,) B )  <->  ( x  e.  RR*  /\  A  <_  x  /\  x  <  B
) ) )
54biimpa 492 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  x  e.  ( A [,) B ) )  ->  ( x  e. 
RR*  /\  A  <_  x  /\  x  <  B
) )
65simp1d 1042 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  x  e.  ( A [,) B ) )  ->  x  e.  RR* )
7 simplr 770 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  x  e.  ( A [,) B ) )  ->  B  e.  RR* )
85simp3d 1044 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  x  e.  ( A [,) B ) )  ->  x  <  B
)
9 xrltne 11483 . . . . . . . . 9  |-  ( ( x  e.  RR*  /\  B  e.  RR*  /\  x  < 
B )  ->  B  =/=  x )
106, 7, 8, 9syl3anc 1292 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  x  e.  ( A [,) B ) )  ->  B  =/=  x
)
1110necomd 2698 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  x  e.  ( A [,) B ) )  ->  x  =/=  B
)
1211neneqd 2648 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  x  e.  ( A [,) B ) )  ->  -.  x  =  B )
13 elsn 3973 . . . . . 6  |-  ( x  e.  { B }  <->  x  =  B )
1412, 13sylnibr 312 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  x  e.  ( A [,) B ) )  ->  -.  x  e.  { B } )
153, 14eldifd 3401 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  x  e.  ( A [,) B ) )  ->  x  e.  ( ( A [,] B
)  \  { B } ) )
1615ex 441 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
x  e.  ( A [,) B )  ->  x  e.  ( ( A [,] B )  \  { B } ) ) )
1716ssrdv 3424 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A [,) B )  C_  ( ( A [,] B )  \  { B } ) )
18 simpll 768 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  x  e.  (
( A [,] B
)  \  { B } ) )  ->  A  e.  RR* )
19 simplr 770 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  x  e.  (
( A [,] B
)  \  { B } ) )  ->  B  e.  RR* )
20 eldifi 3544 . . . . . . 7  |-  ( x  e.  ( ( A [,] B )  \  { B } )  ->  x  e.  ( A [,] B ) )
21 eliccxr 37708 . . . . . . 7  |-  ( x  e.  ( A [,] B )  ->  x  e.  RR* )
2220, 21syl 17 . . . . . 6  |-  ( x  e.  ( ( A [,] B )  \  { B } )  ->  x  e.  RR* )
2322adantl 473 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  x  e.  (
( A [,] B
)  \  { B } ) )  ->  x  e.  RR* )
2420adantl 473 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  x  e.  (
( A [,] B
)  \  { B } ) )  ->  x  e.  ( A [,] B ) )
25 elicc1 11705 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
x  e.  ( A [,] B )  <->  ( x  e.  RR*  /\  A  <_  x  /\  x  <_  B
) ) )
2625adantr 472 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  x  e.  (
( A [,] B
)  \  { B } ) )  -> 
( x  e.  ( A [,] B )  <-> 
( x  e.  RR*  /\  A  <_  x  /\  x  <_  B ) ) )
2724, 26mpbid 215 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  x  e.  (
( A [,] B
)  \  { B } ) )  -> 
( x  e.  RR*  /\  A  <_  x  /\  x  <_  B ) )
2827simp2d 1043 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  x  e.  (
( A [,] B
)  \  { B } ) )  ->  A  <_  x )
29 eldifsni 4089 . . . . . . . 8  |-  ( x  e.  ( ( A [,] B )  \  { B } )  ->  x  =/=  B )
3029necomd 2698 . . . . . . 7  |-  ( x  e.  ( ( A [,] B )  \  { B } )  ->  B  =/=  x )
3130adantl 473 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  x  e.  (
( A [,] B
)  \  { B } ) )  ->  B  =/=  x )
3227simp3d 1044 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  x  e.  (
( A [,] B
)  \  { B } ) )  ->  x  <_  B )
33 xrleltne 11467 . . . . . . 7  |-  ( ( x  e.  RR*  /\  B  e.  RR*  /\  x  <_  B )  ->  (
x  <  B  <->  B  =/=  x ) )
3423, 19, 32, 33syl3anc 1292 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  x  e.  (
( A [,] B
)  \  { B } ) )  -> 
( x  <  B  <->  B  =/=  x ) )
3531, 34mpbird 240 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  x  e.  (
( A [,] B
)  \  { B } ) )  ->  x  <  B )
3618, 19, 23, 28, 35elicod 11710 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  x  e.  (
( A [,] B
)  \  { B } ) )  ->  x  e.  ( A [,) B ) )
3736ex 441 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
x  e.  ( ( A [,] B ) 
\  { B }
)  ->  x  e.  ( A [,) B ) ) )
3837ssrdv 3424 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A [,] B
)  \  { B } )  C_  ( A [,) B ) )
3917, 38eqssd 3435 1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A [,) B )  =  ( ( A [,] B )  \  { B } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641    \ cdif 3387    C_ wss 3390   {csn 3959   class class class wbr 4395  (class class class)co 6308   RR*cxr 9692    < clt 9693    <_ cle 9694   [,)cico 11662   [,]cicc 11663
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-pre-lttri 9631  ax-pre-lttrn 9632
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-po 4760  df-so 4761  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-1st 6812  df-2nd 6813  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-ico 11666  df-icc 11667
This theorem is referenced by: (None)
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