Theorem iccss2 9568

Description: Condition for a closed interval to be a subset of another closed interval. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 28-Apr-2015.)

Assertion

Ref	Expression
iccss2	|- ( ( C e. ( A [,] B ) /\ D e. ( A [,] B ) ) -> ( C [,] D ) C_ ( A [,] B ) )

Proof of Theorem iccss2

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

Step	Hyp	Ref	Expression
1		df-icc 9519	. . . . . 6 |- [,] = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x <_ z /\ z <_ y ) } )
2	1	elixx3g 9525	. . . . 5 |- ( C e. ( A [,] B ) <-> ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A <_ C /\ C <_ B ) ) )
3	2	simplbi 270	. . . 4 |- ( C e. ( A [,] B ) -> ( A e. RR* /\ B e. RR* /\ C e. RR* ) )
4	3	adantr 272	. . 3 |- ( ( C e. ( A [,] B ) /\ D e. ( A [,] B ) ) -> ( A e. RR* /\ B e. RR* /\ C e. RR* ) )
5	4	simp1d 961	. 2 |- ( ( C e. ( A [,] B ) /\ D e. ( A [,] B ) ) -> A e. RR* )
6	4	simp2d 962	. 2 |- ( ( C e. ( A [,] B ) /\ D e. ( A [,] B ) ) -> B e. RR* )
7	2	simprbi 271	. . . 4 |- ( C e. ( A [,] B ) -> ( A <_ C /\ C <_ B ) )
8	7	adantr 272	. . 3 |- ( ( C e. ( A [,] B ) /\ D e. ( A [,] B ) ) -> ( A <_ C /\ C <_ B ) )
9	8	simpld 111	. 2 |- ( ( C e. ( A [,] B ) /\ D e. ( A [,] B ) ) -> A <_ C )
10	1	elixx3g 9525	. . . . 5 |- ( D e. ( A [,] B ) <-> ( ( A e. RR* /\ B e. RR* /\ D e. RR* ) /\ ( A <_ D /\ D <_ B ) ) )
11	10	simprbi 271	. . . 4 |- ( D e. ( A [,] B ) -> ( A <_ D /\ D <_ B ) )
12	11	simprd 113	. . 3 |- ( D e. ( A [,] B ) -> D <_ B )
13	12	adantl 273	. 2 |- ( ( C e. ( A [,] B ) /\ D e. ( A [,] B ) ) -> D <_ B )
14		xrletr 9432	. . 3 |- ( ( A e. RR* /\ C e. RR* /\ w e. RR* ) -> ( ( A <_ C /\ C <_ w ) -> A <_ w ) )
15		xrletr 9432	. . 3 |- ( ( w e. RR* /\ D e. RR* /\ B e. RR* ) -> ( ( w <_ D /\ D <_ B ) -> w <_ B ) )
16	1, 1, 14, 15	ixxss12 9530	. 2 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( A <_ C /\ D <_ B ) ) -> ( C [,] D ) C_ ( A [,] B ) )
17	5, 6, 9, 13, 16	syl22anc 1185	1 |- ( ( C e. ( A [,] B ) /\ D e. ( A [,] B ) ) -> ( C [,] D ) C_ ( A [,] B ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 930   e. wcel 1448   C_ wss 3021   class class class wbr 3875  (class class class)co 5706   RR*cxr 7671   <_ cle 7673   [,]cicc 9515

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-cnex 7586  ax-resscn 7587  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609

This theorem depends on definitions:  df-bi 116  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-rab 2384  df-v 2643  df-sbc 2863  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-id 4153  df-po 4156  df-iso 4157  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-iota 5024  df-fun 5061  df-fv 5067  df-ov 5709  df-oprab 5710  df-mpo 5711  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-icc 9519

This theorem is referenced by: (None)