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Theorem iocinioc2 29541
Description: Intersection between two open-below, closed-above intervals sharing the same upper bound. (Contributed by Thierry Arnoux, 7-Aug-2017.)
Assertion
Ref Expression
iocinioc2 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A <_ B ) -> ( ( A (,] C ) i^i ( B (,] C ) ) = ( B (,] C ) )
Proof of Theorem iocinioc2
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 elin 3796 . . . 4 |- ( x e. ( ( A (,] C ) i^i ( B (,] C ) ) <-> ( x e. ( A (,] C ) /\ x e. ( B (,] C ) ) )
2 simpl1 1064 . . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A <_ B ) -> A e. RR* )
3 simpl3 1066 . . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A <_ B ) -> C e. RR* )
4 elioc1 12217 . . . . . . 7 |- ( ( A e. RR* /\ C e. RR* ) -> ( x e. ( A (,] C ) <-> ( x e. RR* /\ A < x /\ x <_ C ) ) )
52, 3, 4syl2anc 693 . . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A <_ B ) -> ( x e. ( A (,] C ) <-> ( x e. RR* /\ A < x /\ x <_ C ) ) )
6 simpl2 1065 . . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A <_ B ) -> B e. RR* )
7 elioc1 12217 . . . . . . 7 |- ( ( B e. RR* /\ C e. RR* ) -> ( x e. ( B (,] C ) <-> ( x e. RR* /\ B < x /\ x <_ C ) ) )
86, 3, 7syl2anc 693 . . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A <_ B ) -> ( x e. ( B (,] C ) <-> ( x e. RR* /\ B < x /\ x <_ C ) ) )
95, 8anbi12d 747 . . . . 5 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A <_ B ) -> ( ( x e. ( A (,] C ) /\ x e. ( B (,] C ) ) <-> ( ( x e. RR* /\ A < x /\ x <_ C ) /\ ( x e. RR* /\ B < x /\ x <_ C ) ) ) )
10 simp31 1097 . . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A <_ B /\ ( x e. RR* /\ B < x /\ x <_ C ) ) -> x e. RR* )
1123adant3 1081 . . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A <_ B /\ ( x e. RR* /\ B < x /\ x <_ C ) ) -> A e. RR* )
1263adant3 1081 . . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A <_ B /\ ( x e. RR* /\ B < x /\ x <_ C ) ) -> B e. RR* )
13 simp2 1062 . . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A <_ B /\ ( x e. RR* /\ B < x /\ x <_ C ) ) -> A <_ B )
14 simp32 1098 . . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A <_ B /\ ( x e. RR* /\ B < x /\ x <_ C ) ) -> B < x )
1511, 12, 10, 13, 14xrlelttrd 11991 . . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A <_ B /\ ( x e. RR* /\ B < x /\ x <_ C ) ) -> A < x )
16 simp33 1099 . . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A <_ B /\ ( x e. RR* /\ B < x /\ x <_ C ) ) -> x <_ C )
1710, 15, 163jca 1242 . . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A <_ B /\ ( x e. RR* /\ B < x /\ x <_ C ) ) -> ( x e. RR* /\ A < x /\ x <_ C ) )
18173expia 1267 . . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A <_ B ) -> ( ( x e. RR* /\ B < x /\ x <_ C ) -> ( x e. RR* /\ A < x /\ x <_ C ) ) )
1918pm4.71rd 667 . . . . 5 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A <_ B ) -> ( ( x e. RR* /\ B < x /\ x <_ C ) <-> ( ( x e. RR* /\ A < x /\ x <_ C ) /\ ( x e. RR* /\ B < x /\ x <_ C ) ) ) )
209, 19bitr4d 271 . . . 4 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A <_ B ) -> ( ( x e. ( A (,] C ) /\ x e. ( B (,] C ) ) <-> ( x e. RR* /\ B < x /\ x <_ C ) ) )
211, 20syl5bb 272 . . 3 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A <_ B ) -> ( x e. ( ( A (,] C ) i^i ( B (,] C ) ) <-> ( x e. RR* /\ B < x /\ x <_ C ) ) )
2221, 8bitr4d 271 . 2 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A <_ B ) -> ( x e. ( ( A (,] C ) i^i ( B (,] C ) ) <-> x e. ( B (,] C ) ) )
2322eqrdv 2620 1 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A <_ B ) -> ( ( A (,] C ) i^i ( B (,] C ) ) = ( B (,] C ) )
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   class class class wbr 4653  (class class class)co 6650   RR*cxr 10073   < clt 10074   <_ cle 10075   (,]cioc 12176
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
This theorem is referenced by:  iocinif  29543
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