Theorem ioodisj 10227

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 536	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) ) /\ B <_ C ) -> B e. RR* )
2		iooss1 10150	. . . . . 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 10193	. . . . 5 |- ( B (,) D ) C_ ( B [,] D )
5	3, 4	sstrdi 3239	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) ) /\ B <_ C ) -> ( C (,) D ) C_ ( B [,] D ) )
6		sslin 3433	. . . 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 535	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) ) /\ B <_ C ) -> A e. RR* )
9		simplrr 538	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) ) /\ B <_ C ) -> D e. RR* )
10		df-ioo 10126	. . . . 5 |- (,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x < z /\ z < y ) } )
11		df-icc 10129	. . . . 5 |- [,] = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x <_ z /\ z <_ y ) } )
12		xrlenlt 8243	. . . . 5 |- ( ( B e. RR* /\ w e. RR* ) -> ( B <_ w <-> -. w < B ) )
13	10, 11, 12	ixxdisj 10137	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ D e. RR* ) -> ( ( A (,) B ) i^i ( B [,] D ) ) = (/) )
14	8, 1, 9, 13	syl3anc 1273	. . 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 3265	. 2 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) ) /\ B <_ C ) -> ( ( A (,) B ) i^i ( C (,) D ) ) C_ (/) )
16		ss0 3535	. 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 1397   e. wcel 2202   i^i cin 3199   C_ wss 3200   (/)c0 3494   class class class wbr 4088  (class class class)co 6017   RR*cxr 8212   < clt 8213   <_ cle 8214   (,)cioo 10122   [,]cicc 10125

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-po 4393  df-iso 4394  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-ioo 10126  df-icc 10129

This theorem is referenced by: (None)