Theorem iccss2 9946

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 9897	. . . . . 6 |- [,] = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x <_ z /\ z <_ y ) } )
2	1	elixx3g 9903	. . . . 5 |- ( C e. ( A [,] B ) <-> ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A <_ C /\ C <_ B ) ) )
3	2	simplbi 274	. . . 4 |- ( C e. ( A [,] B ) -> ( A e. RR* /\ B e. RR* /\ C e. RR* ) )
4	3	adantr 276	. . 3 |- ( ( C e. ( A [,] B ) /\ D e. ( A [,] B ) ) -> ( A e. RR* /\ B e. RR* /\ C e. RR* ) )
5	4	simp1d 1009	. 2 |- ( ( C e. ( A [,] B ) /\ D e. ( A [,] B ) ) -> A e. RR* )
6	4	simp2d 1010	. 2 |- ( ( C e. ( A [,] B ) /\ D e. ( A [,] B ) ) -> B e. RR* )
7	2	simprbi 275	. . . 4 |- ( C e. ( A [,] B ) -> ( A <_ C /\ C <_ B ) )
8	7	adantr 276	. . 3 |- ( ( C e. ( A [,] B ) /\ D e. ( A [,] B ) ) -> ( A <_ C /\ C <_ B ) )
9	8	simpld 112	. 2 |- ( ( C e. ( A [,] B ) /\ D e. ( A [,] B ) ) -> A <_ C )
10	1	elixx3g 9903	. . . . 5 |- ( D e. ( A [,] B ) <-> ( ( A e. RR* /\ B e. RR* /\ D e. RR* ) /\ ( A <_ D /\ D <_ B ) ) )
11	10	simprbi 275	. . . 4 |- ( D e. ( A [,] B ) -> ( A <_ D /\ D <_ B ) )
12	11	simprd 114	. . 3 |- ( D e. ( A [,] B ) -> D <_ B )
13	12	adantl 277	. 2 |- ( ( C e. ( A [,] B ) /\ D e. ( A [,] B ) ) -> D <_ B )
14		xrletr 9810	. . 3 |- ( ( A e. RR* /\ C e. RR* /\ w e. RR* ) -> ( ( A <_ C /\ C <_ w ) -> A <_ w ) )
15		xrletr 9810	. . 3 |- ( ( w e. RR* /\ D e. RR* /\ B e. RR* ) -> ( ( w <_ D /\ D <_ B ) -> w <_ B ) )
16	1, 1, 14, 15	ixxss12 9908	. 2 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( A <_ C /\ D <_ B ) ) -> ( C [,] D ) C_ ( A [,] B ) )
17	5, 6, 9, 13, 16	syl22anc 1239	1 |- ( ( C e. ( A [,] B ) /\ D e. ( A [,] B ) ) -> ( C [,] D ) C_ ( A [,] B ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 978   e. wcel 2148   C_ wss 3131   class class class wbr 4005  (class class class)co 5877   RR*cxr 7993   <_ cle 7995   [,]cicc 9893

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 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927

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 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-id 4295  df-po 4298  df-iso 4299  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-icc 9897

This theorem is referenced by: (None)