Theorem icodisj 10113

Description: End-to-end closed-below, open-above real intervals are disjoint. (Contributed by Mario Carneiro, 16-Jun-2014.)

Assertion

Ref	Expression
icodisj	|- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( A [,) B ) i^i ( B [,) C ) ) = (/) )

Proof of Theorem icodisj

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elin 3355	. . . 4 |- ( x e. ( ( A [,) B ) i^i ( B [,) C ) ) <-> ( x e. ( A [,) B ) /\ x e. ( B [,) C ) ) )
2		elico1 10044	. . . . . . . . . 10 |- ( ( A e. RR* /\ B e. RR* ) -> ( x e. ( A [,) B ) <-> ( x e. RR* /\ A <_ x /\ x < B ) ) )
3	2	3adant3 1019	. . . . . . . . 9 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( x e. ( A [,) B ) <-> ( x e. RR* /\ A <_ x /\ x < B ) ) )
4	3	biimpa 296	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ x e. ( A [,) B ) ) -> ( x e. RR* /\ A <_ x /\ x < B ) )
5	4	simp3d 1013	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ x e. ( A [,) B ) ) -> x < B )
6	5	adantrr 479	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. ( A [,) B ) /\ x e. ( B [,) C ) ) ) -> x < B )
7		elico1 10044	. . . . . . . . . . 11 |- ( ( B e. RR* /\ C e. RR* ) -> ( x e. ( B [,) C ) <-> ( x e. RR* /\ B <_ x /\ x < C ) ) )
8	7	3adant1 1017	. . . . . . . . . 10 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( x e. ( B [,) C ) <-> ( x e. RR* /\ B <_ x /\ x < C ) ) )
9	8	biimpa 296	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ x e. ( B [,) C ) ) -> ( x e. RR* /\ B <_ x /\ x < C ) )
10	9	simp2d 1012	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ x e. ( B [,) C ) ) -> B <_ x )
11		simpl2 1003	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ x e. ( B [,) C ) ) -> B e. RR* )
12	9	simp1d 1011	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ x e. ( B [,) C ) ) -> x e. RR* )
13		xrlenlt 8136	. . . . . . . . 9 |- ( ( B e. RR* /\ x e. RR* ) -> ( B <_ x <-> -. x < B ) )
14	11, 12, 13	syl2anc 411	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ x e. ( B [,) C ) ) -> ( B <_ x <-> -. x < B ) )
15	10, 14	mpbid 147	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ x e. ( B [,) C ) ) -> -. x < B )
16	15	adantrl 478	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. ( A [,) B ) /\ x e. ( B [,) C ) ) ) -> -. x < B )
17	6, 16	pm2.65da 662	. . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> -. ( x e. ( A [,) B ) /\ x e. ( B [,) C ) ) )
18	17	pm2.21d 620	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( x e. ( A [,) B ) /\ x e. ( B [,) C ) ) -> x e. (/) ) )
19	1, 18	biimtrid 152	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( x e. ( ( A [,) B ) i^i ( B [,) C ) ) -> x e. (/) ) )
20	19	ssrdv 3198	. 2 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( A [,) B ) i^i ( B [,) C ) ) C_ (/) )
21		ss0 3500	. 2 |- ( ( ( A [,) B ) i^i ( B [,) C ) ) C_ (/) -> ( ( A [,) B ) i^i ( B [,) C ) ) = (/) )
22	20, 21	syl 14	1 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( A [,) B ) i^i ( B [,) C ) ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1372   e. wcel 2175   i^i cin 3164   C_ wss 3165   (/)c0 3459   class class class wbr 4043  (class class class)co 5943   RR*cxr 8105   < clt 8106   <_ cle 8107   [,)cico 10011

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-cnex 8015  ax-resscn 8016

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-rab 2492  df-v 2773  df-sbc 2998  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-iota 5231  df-fun 5272  df-fv 5278  df-ov 5946  df-oprab 5947  df-mpo 5948  df-pnf 8108  df-mnf 8109  df-xr 8110  df-le 8112  df-ico 10015

This theorem is referenced by: (None)