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Theorem icoiccdif 37619
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 11718 . . . . . . 7  |-  ( A [,) B )  C_  ( A [,] B )
21a1i 11 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A [,) B )  C_  ( A [,] B ) )
32sselda 3431 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  x  e.  ( A [,) B ) )  ->  x  e.  ( A [,] B ) )
4 elico1 11676 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
x  e.  ( A [,) B )  <->  ( x  e.  RR*  /\  A  <_  x  /\  x  <  B
) ) )
54biimpa 487 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  x  e.  ( A [,) B ) )  ->  ( x  e. 
RR*  /\  A  <_  x  /\  x  <  B
) )
65simp1d 1019 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  x  e.  ( A [,) B ) )  ->  x  e.  RR* )
7 simplr 761 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  x  e.  ( A [,) B ) )  ->  B  e.  RR* )
85simp3d 1021 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  x  e.  ( A [,) B ) )  ->  x  <  B
)
9 xrltne 11457 . . . . . . . . 9  |-  ( ( x  e.  RR*  /\  B  e.  RR*  /\  x  < 
B )  ->  B  =/=  x )
106, 7, 8, 9syl3anc 1267 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  x  e.  ( A [,) B ) )  ->  B  =/=  x
)
1110necomd 2678 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  x  e.  ( A [,) B ) )  ->  x  =/=  B
)
1211neneqd 2628 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  x  e.  ( A [,) B ) )  ->  -.  x  =  B )
13 elsn 3981 . . . . . 6  |-  ( x  e.  { B }  <->  x  =  B )
1412, 13sylnibr 307 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  x  e.  ( A [,) B ) )  ->  -.  x  e.  { B } )
153, 14eldifd 3414 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  x  e.  ( A [,) B ) )  ->  x  e.  ( ( A [,] B
)  \  { B } ) )
1615ex 436 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
x  e.  ( A [,) B )  ->  x  e.  ( ( A [,] B )  \  { B } ) ) )
1716ssrdv 3437 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A [,) B )  C_  ( ( A [,] B )  \  { B } ) )
18 simpll 759 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  x  e.  (
( A [,] B
)  \  { B } ) )  ->  A  e.  RR* )
19 simplr 761 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  x  e.  (
( A [,] B
)  \  { B } ) )  ->  B  e.  RR* )
20 eldifi 3554 . . . . . . 7  |-  ( x  e.  ( ( A [,] B )  \  { B } )  ->  x  e.  ( A [,] B ) )
21 eliccxr 37606 . . . . . . 7  |-  ( x  e.  ( A [,] B )  ->  x  e.  RR* )
2220, 21syl 17 . . . . . 6  |-  ( x  e.  ( ( A [,] B )  \  { B } )  ->  x  e.  RR* )
2322adantl 468 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  x  e.  (
( A [,] B
)  \  { B } ) )  ->  x  e.  RR* )
2420adantl 468 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  x  e.  (
( A [,] B
)  \  { B } ) )  ->  x  e.  ( A [,] B ) )
25 elicc1 11677 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
x  e.  ( A [,] B )  <->  ( x  e.  RR*  /\  A  <_  x  /\  x  <_  B
) ) )
2625adantr 467 . . . . . . 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 214 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  x  e.  (
( A [,] B
)  \  { B } ) )  -> 
( x  e.  RR*  /\  A  <_  x  /\  x  <_  B ) )
2827simp2d 1020 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  x  e.  (
( A [,] B
)  \  { B } ) )  ->  A  <_  x )
29 eldifsni 4097 . . . . . . . 8  |-  ( x  e.  ( ( A [,] B )  \  { B } )  ->  x  =/=  B )
3029necomd 2678 . . . . . . 7  |-  ( x  e.  ( ( A [,] B )  \  { B } )  ->  B  =/=  x )
3130adantl 468 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  x  e.  (
( A [,] B
)  \  { B } ) )  ->  B  =/=  x )
3227simp3d 1021 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  x  e.  (
( A [,] B
)  \  { B } ) )  ->  x  <_  B )
33 xrleltne 11441 . . . . . . 7  |-  ( ( x  e.  RR*  /\  B  e.  RR*  /\  x  <_  B )  ->  (
x  <  B  <->  B  =/=  x ) )
3423, 19, 32, 33syl3anc 1267 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  x  e.  (
( A [,] B
)  \  { B } ) )  -> 
( x  <  B  <->  B  =/=  x ) )
3531, 34mpbird 236 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  x  e.  (
( A [,] B
)  \  { B } ) )  ->  x  <  B )
3618, 19, 23, 28, 35elicod 11682 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  x  e.  (
( A [,] B
)  \  { B } ) )  ->  x  e.  ( A [,) B ) )
3736ex 436 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
x  e.  ( ( A [,] B ) 
\  { B }
)  ->  x  e.  ( A [,) B ) ) )
3837ssrdv 3437 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A [,] B
)  \  { B } )  C_  ( A [,) B ) )
3917, 38eqssd 3448 1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A [,) B )  =  ( ( A [,] B )  \  { B } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 984    = wceq 1443    e. wcel 1886    =/= wne 2621    \ cdif 3400    C_ wss 3403   {csn 3967   class class class wbr 4401  (class class class)co 6288   RR*cxr 9671    < clt 9672    <_ cle 9673   [,)cico 11634   [,]cicc 11635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-pre-lttri 9610  ax-pre-lttrn 9611
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-po 4754  df-so 4755  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-1st 6790  df-2nd 6791  df-er 7360  df-en 7567  df-dom 7568  df-sdom 7569  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-ico 11638  df-icc 11639
This theorem is referenced by: (None)
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