Theorem ioodisj 9991

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 534	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) ) /\ B <_ C ) -> B e. RR* )
2		iooss1 9914	. . . . . 6 |- ( ( B e. RR* /\ B <_ C ) -> ( C (,) D ) C_ ( B (,) D ) )
3	1, 2	sylancom 420	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) ) /\ B <_ C ) -> ( C (,) D ) C_ ( B (,) D ) )
4		ioossicc 9957	. . . . 5 |- ( B (,) D ) C_ ( B [,] D )
5	3, 4	sstrdi 3167	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) ) /\ B <_ C ) -> ( C (,) D ) C_ ( B [,] D ) )
6		sslin 3361	. . . 4 |- ( ( C (,) D ) C_ ( B [,] D ) -> ( ( A (,) B ) i^i ( C (,) D ) ) C_ ( ( A (,) B ) i^i ( B [,] D ) ) )
7	5, 6	syl 14	. . 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 533	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) ) /\ B <_ C ) -> A e. RR* )
9		simplrr 536	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) ) /\ B <_ C ) -> D e. RR* )
10		df-ioo 9890	. . . . 5 |- (,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x < z /\ z < y ) } )
11		df-icc 9893	. . . . 5 |- [,] = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x <_ z /\ z <_ y ) } )
12		xrlenlt 8020	. . . . 5 |- ( ( B e. RR* /\ w e. RR* ) -> ( B <_ w <-> -. w < B ) )
13	10, 11, 12	ixxdisj 9901	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ D e. RR* ) -> ( ( A (,) B ) i^i ( B [,] D ) ) = (/) )
14	8, 1, 9, 13	syl3anc 1238	. . 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 3193	. 2 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) ) /\ B <_ C ) -> ( ( A (,) B ) i^i ( C (,) D ) ) C_ (/) )
16		ss0 3463	. 2 |- ( ( ( A (,) B ) i^i ( C (,) D ) ) C_ (/) -> ( ( A (,) B ) i^i ( C (,) D ) ) = (/) )
17	15, 16	syl 14	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 104   = wceq 1353   e. wcel 2148   i^i cin 3128   C_ wss 3129   (/)c0 3422   class class class wbr 4003  (class class class)co 5874   RR*cxr 7989   < clt 7990   <_ cle 7991   (,)cioo 9886   [,]cicc 9889

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-id 4293  df-po 4296  df-iso 4297  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-iota 5178  df-fun 5218  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-pnf 7992  df-mnf 7993  df-xr 7994  df-ltxr 7995  df-le 7996  df-ioo 9890  df-icc 9893

This theorem is referenced by: (None)