Theorem icodisj 12297

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 3796	. . . 4 |- ( x e. ( ( A [,) B ) i^i ( B [,) C ) ) <-> ( x e. ( A [,) B ) /\ x e. ( B [,) C ) ) )
2		elico1 12218	. . . . . . . . . 10 |- ( ( A e. RR* /\ B e. RR* ) -> ( x e. ( A [,) B ) <-> ( x e. RR* /\ A <_ x /\ x < B ) ) )
3	2	3adant3 1081	. . . . . . . . 9 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( x e. ( A [,) B ) <-> ( x e. RR* /\ A <_ x /\ x < B ) ) )
4	3	biimpa 501	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ x e. ( A [,) B ) ) -> ( x e. RR* /\ A <_ x /\ x < B ) )
5	4	simp3d 1075	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ x e. ( A [,) B ) ) -> x < B )
6	5	adantrr 753	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. ( A [,) B ) /\ x e. ( B [,) C ) ) ) -> x < B )
7		elico1 12218	. . . . . . . . . . 11 |- ( ( B e. RR* /\ C e. RR* ) -> ( x e. ( B [,) C ) <-> ( x e. RR* /\ B <_ x /\ x < C ) ) )
8	7	3adant1 1079	. . . . . . . . . 10 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( x e. ( B [,) C ) <-> ( x e. RR* /\ B <_ x /\ x < C ) ) )
9	8	biimpa 501	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ x e. ( B [,) C ) ) -> ( x e. RR* /\ B <_ x /\ x < C ) )
10	9	simp2d 1074	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ x e. ( B [,) C ) ) -> B <_ x )
11		simpl2 1065	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ x e. ( B [,) C ) ) -> B e. RR* )
12	9	simp1d 1073	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ x e. ( B [,) C ) ) -> x e. RR* )
13		xrlenlt 10103	. . . . . . . . 9 |- ( ( B e. RR* /\ x e. RR* ) -> ( B <_ x <-> -. x < B ) )
14	11, 12, 13	syl2anc 693	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ x e. ( B [,) C ) ) -> ( B <_ x <-> -. x < B ) )
15	10, 14	mpbid 222	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ x e. ( B [,) C ) ) -> -. x < B )
16	15	adantrl 752	. . . . . 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 600	. . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> -. ( x e. ( A [,) B ) /\ x e. ( B [,) C ) ) )
18	17	pm2.21d 118	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( x e. ( A [,) B ) /\ x e. ( B [,) C ) ) -> x e. (/) ) )
19	1, 18	syl5bi 232	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( x e. ( ( A [,) B ) i^i ( B [,) C ) ) -> x e. (/) ) )
20	19	ssrdv 3609	. 2 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( A [,) B ) i^i ( B [,) C ) ) C_ (/) )
21		ss0 3974	. 2 |- ( ( ( A [,) B ) i^i ( B [,) C ) ) C_ (/) -> ( ( A [,) B ) i^i ( B [,) C ) ) = (/) )
22	20, 21	syl 17	1 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( A [,) B ) i^i ( B [,) C ) ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   i^i cin 3573   C_ wss 3574   (/)c0 3915   class class class wbr 4653  (class class class)co 6650   RR*cxr 10073   < clt 10074   <_ cle 10075   [,)cico 12177

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-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-xr 10078  df-le 10080  df-ico 12181

This theorem is referenced by:  icombl  23332  difico  29545  chtvalz  30707