Theorem ioodisj 10395

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 540	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) ) /\ B <_ C ) -> B e. RR* )
2		iooss1 10318	. . . . . 6 |- ( ( B e. RR* /\ B <_ C ) -> ( C (,) D ) C_ ( B (,) D ) )
3	1, 2	sylancom 424	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) ) /\ B <_ C ) -> ( C (,) D ) C_ ( B (,) D ) )
4		ioossicc 10361	. . . . 5 |- ( B (,) D ) C_ ( B [,] D )
5	3, 4	sstrdi 3260	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) ) /\ B <_ C ) -> ( C (,) D ) C_ ( B [,] D ) )
6		sslin 3457	. . . 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 539	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) ) /\ B <_ C ) -> A e. RR* )
9		simplrr 542	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) ) /\ B <_ C ) -> D e. RR* )
10		df-ioo 10294	. . . . 5 |- (,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x < z /\ z < y ) } )
11		df-icc 10297	. . . . 5 |- [,] = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x <_ z /\ z <_ y ) } )
12		xrlenlt 8390	. . . . 5 |- ( ( B e. RR* /\ w e. RR* ) -> ( B <_ w <-> -. w < B ) )
13	10, 11, 12	ixxdisj 10305	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ D e. RR* ) -> ( ( A (,) B ) i^i ( B [,] D ) ) = (/) )
14	8, 1, 9, 13	syl3anc 1278	. . 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 3286	. 2 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) ) /\ B <_ C ) -> ( ( A (,) B ) i^i ( C (,) D ) ) C_ (/) )
16		ss0 3563	. 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 ) ) = (/) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   i^i cin 3219   C_ wss 3220   (/)c0 3520   class class class wbr 4130  (class class class)co 6085   RR*cxr 8359   < clt 8360   <_ cle 8361   (,)cioo 10290   [,]cicc 10293

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-id 4438  df-po 4441  df-iso 4442  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-iota 5337  df-fun 5379  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-ioo 10294  df-icc 10297

This theorem is used by: (None)