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Theorem iocinico 37797
Description: The intersection of two sets that meet at a point is that point. (Contributed by Jon Pennant, 12-Jun-2019.)
Assertion
Ref Expression
iocinico |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < B /\ B < C ) ) -> ( ( A (,] B ) i^i ( B [,) C ) ) = { B } )
Proof of Theorem iocinico
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 df-in 3581 . . . . . 6 |- ( ( A (,] B ) i^i ( B [,) C ) ) = { x | ( x e. ( A (,] B ) /\ x e. ( B [,) C ) ) }
2 elioc1 12217 . . . . . . . . . . . 12 |- ( ( A e. RR* /\ B e. RR* ) -> ( x e. ( A (,] B ) <-> ( x e. RR* /\ A < x /\ x <_ B ) ) )
323adant3 1081 . . . . . . . . . . 11 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( x e. ( A (,] B ) <-> ( x e. RR* /\ A < x /\ x <_ B ) ) )
4 3simpb 1059 . . . . . . . . . . 11 |- ( ( x e. RR* /\ A < x /\ x <_ B ) -> ( x e. RR* /\ x <_ B ) )
53, 4syl6bi 243 . . . . . . . . . 10 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( x e. ( A (,] B ) -> ( x e. RR* /\ x <_ B ) ) )
6 elico1 12218 . . . . . . . . . . . 12 |- ( ( B e. RR* /\ C e. RR* ) -> ( x e. ( B [,) C ) <-> ( x e. RR* /\ B <_ x /\ x < C ) ) )
763adant1 1079 . . . . . . . . . . 11 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( x e. ( B [,) C ) <-> ( x e. RR* /\ B <_ x /\ x < C ) ) )
8 3simpa 1058 . . . . . . . . . . 11 |- ( ( x e. RR* /\ B <_ x /\ x < C ) -> ( x e. RR* /\ B <_ x ) )
97, 8syl6bi 243 . . . . . . . . . 10 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( x e. ( B [,) C ) -> ( x e. RR* /\ B <_ x ) ) )
105, 9anim12d 586 . . . . . . . . 9 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( x e. ( A (,] B ) /\ x e. ( B [,) C ) ) -> ( ( x e. RR* /\ x <_ B ) /\ ( x e. RR* /\ B <_ x ) ) ) )
11 simpll 790 . . . . . . . . . 10 |- ( ( ( x e. RR* /\ x <_ B ) /\ ( x e. RR* /\ B <_ x ) ) -> x e. RR* )
12 simprr 796 . . . . . . . . . 10 |- ( ( ( x e. RR* /\ x <_ B ) /\ ( x e. RR* /\ B <_ x ) ) -> B <_ x )
13 simplr 792 . . . . . . . . . 10 |- ( ( ( x e. RR* /\ x <_ B ) /\ ( x e. RR* /\ B <_ x ) ) -> x <_ B )
1411, 12, 133jca 1242 . . . . . . . . 9 |- ( ( ( x e. RR* /\ x <_ B ) /\ ( x e. RR* /\ B <_ x ) ) -> ( x e. RR* /\ B <_ x /\ x <_ B ) )
1510, 14syl6 35 . . . . . . . 8 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( x e. ( A (,] B ) /\ x e. ( B [,) C ) ) -> ( x e. RR* /\ B <_ x /\ x <_ B ) ) )
16 elicc1 12219 . . . . . . . . . 10 |- ( ( B e. RR* /\ B e. RR* ) -> ( x e. ( B [,] B ) <-> ( x e. RR* /\ B <_ x /\ x <_ B ) ) )
1716anidms 677 . . . . . . . . 9 |- ( B e. RR* -> ( x e. ( B [,] B ) <-> ( x e. RR* /\ B <_ x /\ x <_ B ) ) )
18173ad2ant2 1083 . . . . . . . 8 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( x e. ( B [,] B ) <-> ( x e. RR* /\ B <_ x /\ x <_ B ) ) )
1915, 18sylibrd 249 . . . . . . 7 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( x e. ( A (,] B ) /\ x e. ( B [,) C ) ) -> x e. ( B [,] B ) ) )
2019ss2abdv 3675 . . . . . 6 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> { x | ( x e. ( A (,] B ) /\ x e. ( B [,) C ) ) } C_ { x | x e. ( B [,] B ) } )
211, 20syl5eqss 3649 . . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( A (,] B ) i^i ( B [,) C ) ) C_ { x | x e. ( B [,] B ) } )
22 abid2 2745 . . . . 5 |- { x | x e. ( B [,] B ) } = ( B [,] B )
2321, 22syl6sseq 3651 . . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( A (,] B ) i^i ( B [,) C ) ) C_ ( B [,] B ) )
2423adantr 481 . . 3 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < B /\ B < C ) ) -> ( ( A (,] B ) i^i ( B [,) C ) ) C_ ( B [,] B ) )
25 iccid 12220 . . . . 5 |- ( B e. RR* -> ( B [,] B ) = { B } )
26253ad2ant2 1083 . . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( B [,] B ) = { B } )
2726adantr 481 . . 3 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < B /\ B < C ) ) -> ( B [,] B ) = { B } )
2824, 27sseqtrd 3641 . 2 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < B /\ B < C ) ) -> ( ( A (,] B ) i^i ( B [,) C ) ) C_ { B } )
29 simpl2 1065 . . . . 5 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < B /\ B < C ) ) -> B e. RR* )
30 simprl 794 . . . . 5 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < B /\ B < C ) ) -> A < B )
31 xrleid 11983 . . . . . 6 |- ( B e. RR* -> B <_ B )
3229, 31syl 17 . . . . 5 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < B /\ B < C ) ) -> B <_ B )
33 elioc1 12217 . . . . . . 7 |- ( ( A e. RR* /\ B e. RR* ) -> ( B e. ( A (,] B ) <-> ( B e. RR* /\ A < B /\ B <_ B ) ) )
34333adant3 1081 . . . . . 6 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( B e. ( A (,] B ) <-> ( B e. RR* /\ A < B /\ B <_ B ) ) )
3534adantr 481 . . . . 5 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < B /\ B < C ) ) -> ( B e. ( A (,] B ) <-> ( B e. RR* /\ A < B /\ B <_ B ) ) )
3629, 30, 32, 35mpbir3and 1245 . . . 4 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < B /\ B < C ) ) -> B e. ( A (,] B ) )
37 simprr 796 . . . . 5 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < B /\ B < C ) ) -> B < C )
38 elico1 12218 . . . . . . 7 |- ( ( B e. RR* /\ C e. RR* ) -> ( B e. ( B [,) C ) <-> ( B e. RR* /\ B <_ B /\ B < C ) ) )
39383adant1 1079 . . . . . 6 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( B e. ( B [,) C ) <-> ( B e. RR* /\ B <_ B /\ B < C ) ) )
4039adantr 481 . . . . 5 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < B /\ B < C ) ) -> ( B e. ( B [,) C ) <-> ( B e. RR* /\ B <_ B /\ B < C ) ) )
4129, 32, 37, 40mpbir3and 1245 . . . 4 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < B /\ B < C ) ) -> B e. ( B [,) C ) )
4236, 41elind 3798 . . 3 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < B /\ B < C ) ) -> B e. ( ( A (,] B ) i^i ( B [,) C ) ) )
4342snssd 4340 . 2 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < B /\ B < C ) ) -> { B } C_ ( ( A (,] B ) i^i ( B [,) C ) ) )
4428, 43eqssd 3620 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   {cab 2608   i^i cin 3573   C_ wss 3574   {csn 4177   class class class wbr 4653  (class class class)co 6650   RR*cxr 10073   < clt 10074   <_ cle 10075   (,]cioc 12176   [,)cico 12177   [,]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-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-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-ioc 12180  df-ico 12181  df-icc 12182
This theorem is referenced by: (None)
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