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Theorem iocunico 31419
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 3649 . . 3  |-  ( ( ( A (,) B
)  u.  { B } )  u.  ( B (,) C ) )  =  ( ( ( A (,) B )  u.  ( B (,) C ) )  u. 
{ B } )
2 unundir 3652 . . 3  |-  ( ( ( A (,) B
)  u.  ( B (,) C ) )  u.  { B }
)  =  ( ( ( A (,) B
)  u.  { B } )  u.  (
( B (,) C
)  u.  { B } ) )
3 uncom 3634 . . . 4  |-  ( ( B (,) C )  u.  { B }
)  =  ( { B }  u.  ( B (,) C ) )
43uneq2i 3641 . . 3  |-  ( ( ( A (,) B
)  u.  { B } )  u.  (
( B (,) C
)  u.  { B } ) )  =  ( ( ( A (,) B )  u. 
{ B } )  u.  ( { B }  u.  ( B (,) C ) ) )
51, 2, 43eqtrri 2488 . 2  |-  ( ( ( A (,) B
)  u.  { B } )  u.  ( { B }  u.  ( B (,) C ) ) )  =  ( ( ( A (,) B
)  u.  { B } )  u.  ( B (,) C ) )
6 simpl1 997 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  ->  A  e.  RR* )
7 simpl2 998 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  ->  B  e.  RR* )
8 simprl 754 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  ->  A  <  B )
9 ioounsn 31418 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  (
( A (,) B
)  u.  { B } )  =  ( A (,] B ) )
106, 7, 8, 9syl3anc 1226 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  -> 
( ( A (,) B )  u.  { B } )  =  ( A (,] B ) )
11 simpl3 999 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  ->  C  e.  RR* )
12 simprr 755 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  ->  B  <  C )
13 snunioo 11649 . . . 4  |-  ( ( B  e.  RR*  /\  C  e.  RR*  /\  B  < 
C )  ->  ( { B }  u.  ( B (,) C ) )  =  ( B [,) C ) )
147, 11, 12, 13syl3anc 1226 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  -> 
( { B }  u.  ( B (,) C
) )  =  ( B [,) C ) )
1510, 14uneq12d 3645 . 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 11654 . 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 2519 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823    u. cun 3459   {csn 4016   class class class wbr 4439  (class class class)co 6270   RR*cxr 9616    < clt 9617   (,)cioo 11532   (,]cioc 11533   [,)cico 11534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-pre-lttri 9555  ax-pre-lttrn 9556
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-po 4789  df-so 4790  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-ioo 11536  df-ioc 11537  df-ico 11538  df-icc 11539
This theorem is referenced by: (None)
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