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Theorem iocunico 29729
Description: Split an open interval into two pieces at point B, Co-author TA. (Contributed by Jon Pennant, 8-Jun-2019.)
Assertion
Ref Expression
iocunico  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  -> 
( ( A (,] B )  u.  ( B [,) C ) )  =  ( A (,) C ) )

Proof of Theorem iocunico
StepHypRef Expression
1 un23 3618 . . 3  |-  ( ( ( A (,) B
)  u.  { B } )  u.  ( B (,) C ) )  =  ( ( ( A (,) B )  u.  ( B (,) C ) )  u. 
{ B } )
2 unundir 3621 . . 3  |-  ( ( ( A (,) B
)  u.  ( B (,) C ) )  u.  { B }
)  =  ( ( ( A (,) B
)  u.  { B } )  u.  (
( B (,) C
)  u.  { B } ) )
3 uncom 3603 . . . 4  |-  ( ( B (,) C )  u.  { B }
)  =  ( { B }  u.  ( B (,) C ) )
43uneq2i 3610 . . 3  |-  ( ( ( A (,) B
)  u.  { B } )  u.  (
( B (,) C
)  u.  { B } ) )  =  ( ( ( A (,) B )  u. 
{ B } )  u.  ( { B }  u.  ( B (,) C ) ) )
51, 2, 43eqtrri 2486 . 2  |-  ( ( ( A (,) B
)  u.  { B } )  u.  ( { B }  u.  ( B (,) C ) ) )  =  ( ( ( A (,) B
)  u.  { B } )  u.  ( B (,) C ) )
6 simpl1 991 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  ->  A  e.  RR* )
7 simpl2 992 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  ->  B  e.  RR* )
8 simprl 755 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  ->  A  <  B )
9 ioounsn 29728 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  (
( A (,) B
)  u.  { B } )  =  ( A (,] B ) )
106, 7, 8, 9syl3anc 1219 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  -> 
( ( A (,) B )  u.  { B } )  =  ( A (,] B ) )
11 simpl3 993 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  ->  C  e.  RR* )
12 simprr 756 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  ->  B  <  C )
13 snunioo 11523 . . . 4  |-  ( ( B  e.  RR*  /\  C  e.  RR*  /\  B  < 
C )  ->  ( { B }  u.  ( B (,) C ) )  =  ( B [,) C ) )
147, 11, 12, 13syl3anc 1219 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  -> 
( { B }  u.  ( B (,) C
) )  =  ( B [,) C ) )
1510, 14uneq12d 3614 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  -> 
( ( ( A (,) B )  u. 
{ B } )  u.  ( { B }  u.  ( B (,) C ) ) )  =  ( ( A (,] B )  u.  ( B [,) C
) ) )
16 ioojoin 11528 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  -> 
( ( ( A (,) B )  u. 
{ B } )  u.  ( B (,) C ) )  =  ( A (,) C
) )
175, 15, 163eqtr3a 2517 1  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  -> 
( ( A (,] B )  u.  ( B [,) C ) )  =  ( A (,) C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    u. cun 3429   {csn 3980   class class class wbr 4395  (class class class)co 6195   RR*cxr 9523    < clt 9524   (,)cioo 11406   (,]cioc 11407   [,)cico 11408
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-cnex 9444  ax-resscn 9445  ax-pre-lttri 9462  ax-pre-lttrn 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-po 4744  df-so 4745  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-er 7206  df-en 7416  df-dom 7417  df-sdom 7418  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-ioo 11410  df-ioc 11411  df-ico 11412  df-icc 11413
This theorem is referenced by: (None)
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