Theorem iccss2 11366

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 11307	. . . . . 6 |- [,] = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x <_ z /\ z <_ y ) } )
2	1	elixx3g 11313	. . . . 5 |- ( C e. ( A [,] B ) <-> ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A <_ C /\ C <_ B ) ) )
3	2	simplbi 460	. . . 4 |- ( C e. ( A [,] B ) -> ( A e. RR* /\ B e. RR* /\ C e. RR* ) )
4	3	adantr 465	. . 3 |- ( ( C e. ( A [,] B ) /\ D e. ( A [,] B ) ) -> ( A e. RR* /\ B e. RR* /\ C e. RR* ) )
5	4	simp1d 1000	. 2 |- ( ( C e. ( A [,] B ) /\ D e. ( A [,] B ) ) -> A e. RR* )
6	4	simp2d 1001	. 2 |- ( ( C e. ( A [,] B ) /\ D e. ( A [,] B ) ) -> B e. RR* )
7	2	simprbi 464	. . . 4 |- ( C e. ( A [,] B ) -> ( A <_ C /\ C <_ B ) )
8	7	adantr 465	. . 3 |- ( ( C e. ( A [,] B ) /\ D e. ( A [,] B ) ) -> ( A <_ C /\ C <_ B ) )
9	8	simpld 459	. 2 |- ( ( C e. ( A [,] B ) /\ D e. ( A [,] B ) ) -> A <_ C )
10	1	elixx3g 11313	. . . . 5 |- ( D e. ( A [,] B ) <-> ( ( A e. RR* /\ B e. RR* /\ D e. RR* ) /\ ( A <_ D /\ D <_ B ) ) )
11	10	simprbi 464	. . . 4 |- ( D e. ( A [,] B ) -> ( A <_ D /\ D <_ B ) )
12	11	simprd 463	. . 3 |- ( D e. ( A [,] B ) -> D <_ B )
13	12	adantl 466	. 2 |- ( ( C e. ( A [,] B ) /\ D e. ( A [,] B ) ) -> D <_ B )
14		xrletr 11132	. . 3 |- ( ( A e. RR* /\ C e. RR* /\ w e. RR* ) -> ( ( A <_ C /\ C <_ w ) -> A <_ w ) )
15		xrletr 11132	. . 3 |- ( ( w e. RR* /\ D e. RR* /\ B e. RR* ) -> ( ( w <_ D /\ D <_ B ) -> w <_ B ) )
16	1, 1, 14, 15	ixxss12 11320	. 2 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( A <_ C /\ D <_ B ) ) -> ( C [,] D ) C_ ( A [,] B ) )
17	5, 6, 9, 13, 16	syl22anc 1219	1 |- ( ( C e. ( A [,] B ) /\ D e. ( A [,] B ) ) -> ( C [,] D ) C_ ( A [,] B ) )

Syntax hints:   -> wi 4   /\ wa 369   /\ w3a 965   e. wcel 1756   C_ wss 3328   class class class wbr 4292  (class class class)co 6091   RR*cxr 9417   <_ cle 9419   [,]cicc 11303

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-pre-lttri 9356  ax-pre-lttrn 9357

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-po 4641  df-so 4642  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-1st 6577  df-2nd 6578  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-icc 11307

This theorem is referenced by:  ordtresticc  18827  iccconn  20407  icccvx  20522  oprpiece1res1  20523  oprpiece1res2  20524  pcoass  20596  dvlip  21465  c1liplem1  21468  dvgt0lem1  21474  ftc2ditglem  21517  ttgcontlem1  23131  unitssxrge0  26330  xrge0iifhmeo  26366