Theorem ioodisj 12302

Description: If the upper bound of one open interval is less than or equal to the lower bound of the other, the intervals are disjoint. (Contributed by Jeff Hankins, 13-Jul-2009.)

Assertion

Ref	Expression
ioodisj	|- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) ) /\ B <_ C ) -> ( ( A (,) B ) i^i ( C (,) D ) ) = (/) )

Proof of Theorem ioodisj

Dummy variables x w y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpllr 799	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) ) /\ B <_ C ) -> B e. RR* )
2		iooss1 12210	. . . . . 6 |- ( ( B e. RR* /\ B <_ C ) -> ( C (,) D ) C_ ( B (,) D ) )
3	1, 2	sylancom 701	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) ) /\ B <_ C ) -> ( C (,) D ) C_ ( B (,) D ) )
4		ioossicc 12259	. . . . 5 |- ( B (,) D ) C_ ( B [,] D )
5	3, 4	syl6ss 3615	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) ) /\ B <_ C ) -> ( C (,) D ) C_ ( B [,] D ) )
6		sslin 3839	. . . 4 |- ( ( C (,) D ) C_ ( B [,] D ) -> ( ( A (,) B ) i^i ( C (,) D ) ) C_ ( ( A (,) B ) i^i ( B [,] D ) ) )
7	5, 6	syl 17	. . 3 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) ) /\ B <_ C ) -> ( ( A (,) B ) i^i ( C (,) D ) ) C_ ( ( A (,) B ) i^i ( B [,] D ) ) )
8		simplll 798	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) ) /\ B <_ C ) -> A e. RR* )
9		simplrr 801	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) ) /\ B <_ C ) -> D e. RR* )
10		df-ioo 12179	. . . . 5 |- (,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x < z /\ z < y ) } )
11		df-icc 12182	. . . . 5 |- [,] = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x <_ z /\ z <_ y ) } )
12		xrlenlt 10103	. . . . 5 |- ( ( B e. RR* /\ w e. RR* ) -> ( B <_ w <-> -. w < B ) )
13	10, 11, 12	ixxdisj 12190	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ D e. RR* ) -> ( ( A (,) B ) i^i ( B [,] D ) ) = (/) )
14	8, 1, 9, 13	syl3anc 1326	. . 3 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) ) /\ B <_ C ) -> ( ( A (,) B ) i^i ( B [,] D ) ) = (/) )
15	7, 14	sseqtrd 3641	. 2 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) ) /\ B <_ C ) -> ( ( A (,) B ) i^i ( C (,) D ) ) C_ (/) )
16		ss0 3974	. 2 |- ( ( ( A (,) B ) i^i ( C (,) D ) ) C_ (/) -> ( ( A (,) B ) i^i ( C (,) D ) ) = (/) )
17	15, 16	syl 17	1 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) ) /\ B <_ C ) -> ( ( A (,) B ) i^i ( C (,) D ) ) = (/) )

Syntax hints:   -> wi 4   /\ wa 384   = 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   (,)cioo 12175   [,]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-ioo 12179  df-icc 12182

This theorem is referenced by:  reconnlem1  22629  dyaddisjlem  23363  itgsplitioo  23604