Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  iccintsng Structured version   Visualization version   Unicode version
Theorem iccintsng 39749
Description: Intersection of two adiacent closed intervals is a singleton. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Assertion
Ref Expression
iccintsng |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A <_ B /\ B <_ C ) ) -> ( ( A [,] B ) i^i ( B [,] C ) ) = { B } )
Proof of Theorem iccintsng
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 simpl1 1064 . . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. ( A [,] B ) /\ x e. ( B [,] C ) ) ) -> A e. RR* )
2 simpl2 1065 . . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. ( A [,] B ) /\ x e. ( B [,] C ) ) ) -> B e. RR* )
3 simprl 794 . . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. ( A [,] B ) /\ x e. ( B [,] C ) ) ) -> x e. ( A [,] B ) )
4 iccleub 12229 . . . . . . . 8 |- ( ( A e. RR* /\ B e. RR* /\ x e. ( A [,] B ) ) -> x <_ B )
51, 2, 3, 4syl3anc 1326 . . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. ( A [,] B ) /\ x e. ( B [,] C ) ) ) -> x <_ B )
6 simpl3 1066 . . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. ( A [,] B ) /\ x e. ( B [,] C ) ) ) -> C e. RR* )
7 simprr 796 . . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. ( A [,] B ) /\ x e. ( B [,] C ) ) ) -> x e. ( B [,] C ) )
8 iccgelb 12230 . . . . . . . 8 |- ( ( B e. RR* /\ C e. RR* /\ x e. ( B [,] C ) ) -> B <_ x )
92, 6, 7, 8syl3anc 1326 . . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. ( A [,] B ) /\ x e. ( B [,] C ) ) ) -> B <_ x )
10 eliccxr 39737 . . . . . . . . . 10 |- ( x e. ( A [,] B ) -> x e. RR* )
113, 10syl 17 . . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. ( A [,] B ) /\ x e. ( B [,] C ) ) ) -> x e. RR* )
1211, 2jca 554 . . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. ( A [,] B ) /\ x e. ( B [,] C ) ) ) -> ( x e. RR* /\ B e. RR* ) )
13 xrletri3 11985 . . . . . . . 8 |- ( ( x e. RR* /\ B e. RR* ) -> ( x = B <-> ( x <_ B /\ B <_ x ) ) )
1412, 13syl 17 . . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. ( A [,] B ) /\ x e. ( B [,] C ) ) ) -> ( x = B <-> ( x <_ B /\ B <_ x ) ) )
155, 9, 14mpbir2and 957 . . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. ( A [,] B ) /\ x e. ( B [,] C ) ) ) -> x = B )
1615ex 450 . . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( x e. ( A [,] B ) /\ x e. ( B [,] C ) ) -> x = B ) )
1716adantr 481 . . . 4 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A <_ B /\ B <_ C ) ) -> ( ( x e. ( A [,] B ) /\ x e. ( B [,] C ) ) -> x = B ) )
18 simpll1 1100 . . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A <_ B /\ B <_ C ) ) /\ x = B ) -> A e. RR* )
19 simpll2 1101 . . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A <_ B /\ B <_ C ) ) /\ x = B ) -> B e. RR* )
20 simplrl 800 . . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A <_ B /\ B <_ C ) ) /\ x = B ) -> A <_ B )
21 simpr 477 . . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A <_ B /\ B <_ C ) ) /\ x = B ) -> x = B )
22 simpr 477 . . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* /\ A <_ B ) /\ x = B ) -> x = B )
23 ubicc2 12289 . . . . . . . . 9 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> B e. ( A [,] B ) )
2423adantr 481 . . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* /\ A <_ B ) /\ x = B ) -> B e. ( A [,] B ) )
2522, 24eqeltrd 2701 . . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* /\ A <_ B ) /\ x = B ) -> x e. ( A [,] B ) )
2618, 19, 20, 21, 25syl31anc 1329 . . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A <_ B /\ B <_ C ) ) /\ x = B ) -> x e. ( A [,] B ) )
27 simpll3 1102 . . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A <_ B /\ B <_ C ) ) /\ x = B ) -> C e. RR* )
28 simplrr 801 . . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A <_ B /\ B <_ C ) ) /\ x = B ) -> B <_ C )
29 simpr 477 . . . . . . . 8 |- ( ( ( B e. RR* /\ C e. RR* /\ B <_ C ) /\ x = B ) -> x = B )
30 lbicc2 12288 . . . . . . . . 9 |- ( ( B e. RR* /\ C e. RR* /\ B <_ C ) -> B e. ( B [,] C ) )
3130adantr 481 . . . . . . . 8 |- ( ( ( B e. RR* /\ C e. RR* /\ B <_ C ) /\ x = B ) -> B e. ( B [,] C ) )
3229, 31eqeltrd 2701 . . . . . . 7 |- ( ( ( B e. RR* /\ C e. RR* /\ B <_ C ) /\ x = B ) -> x e. ( B [,] C ) )
3319, 27, 28, 21, 32syl31anc 1329 . . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A <_ B /\ B <_ C ) ) /\ x = B ) -> x e. ( B [,] C ) )
3426, 33jca 554 . . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A <_ B /\ B <_ C ) ) /\ x = B ) -> ( x e. ( A [,] B ) /\ x e. ( B [,] C ) ) )
3534ex 450 . . . 4 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A <_ B /\ B <_ C ) ) -> ( x = B -> ( x e. ( A [,] B ) /\ x e. ( B [,] C ) ) ) )
3617, 35impbid 202 . . 3 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A <_ B /\ B <_ C ) ) -> ( ( x e. ( A [,] B ) /\ x e. ( B [,] C ) ) <-> x = B ) )
37 elin 3796 . . 3 |- ( x e. ( ( A [,] B ) i^i ( B [,] C ) ) <-> ( x e. ( A [,] B ) /\ x e. ( B [,] C ) ) )
38 velsn 4193 . . 3 |- ( x e. { B } <-> x = B )
3936, 37, 383bitr4g 303 . 2 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A <_ B /\ B <_ C ) ) -> ( x e. ( ( A [,] B ) i^i ( B [,] C ) ) <-> x e. { B } ) )
4039eqrdv 2620 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 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   i^i cin 3573   {csn 4177   class class class wbr 4653  (class class class)co 6650   RR*cxr 10073   <_ cle 10075   [,]cicc 12178
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-pre-lttri 10010  ax-pre-lttrn 10011
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-icc 12182
This theorem is referenced by:  iblspltprt  40189
  Copyright terms: Public domain W3C validator