Theorem ioodisj 9806

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 524	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) ) /\ B <_ C ) -> B e. RR* )
2		iooss1 9729	. . . . . 6 |- ( ( B e. RR* /\ B <_ C ) -> ( C (,) D ) C_ ( B (,) D ) )
3	1, 2	sylancom 417	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) ) /\ B <_ C ) -> ( C (,) D ) C_ ( B (,) D ) )
4		ioossicc 9772	. . . . 5 |- ( B (,) D ) C_ ( B [,] D )
5	3, 4	sstrdi 3114	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) ) /\ B <_ C ) -> ( C (,) D ) C_ ( B [,] D ) )
6		sslin 3307	. . . 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 523	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) ) /\ B <_ C ) -> A e. RR* )
9		simplrr 526	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) ) /\ B <_ C ) -> D e. RR* )
10		df-ioo 9705	. . . . 5 |- (,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x < z /\ z < y ) } )
11		df-icc 9708	. . . . 5 |- [,] = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x <_ z /\ z <_ y ) } )
12		xrlenlt 7853	. . . . 5 |- ( ( B e. RR* /\ w e. RR* ) -> ( B <_ w <-> -. w < B ) )
13	10, 11, 12	ixxdisj 9716	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ D e. RR* ) -> ( ( A (,) B ) i^i ( B [,] D ) ) = (/) )
14	8, 1, 9, 13	syl3anc 1217	. . 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 3140	. 2 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) ) /\ B <_ C ) -> ( ( A (,) B ) i^i ( C (,) D ) ) C_ (/) )
16		ss0 3408	. 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 103   = wceq 1332   e. wcel 1481   i^i cin 3075   C_ wss 3076   (/)c0 3368   class class class wbr 3937  (class class class)co 5782   RR*cxr 7823   < clt 7824   <_ cle 7825   (,)cioo 9701   [,]cicc 9704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-cnex 7735  ax-resscn 7736  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758

This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2914  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-id 4223  df-po 4226  df-iso 4227  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-iota 5096  df-fun 5133  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-ioo 9705  df-icc 9708

This theorem is referenced by: (None)