Theorem iccss2 10277

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 10228	. . . . . 6 |- [,] = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x <_ z /\ z <_ y ) } )
2	1	elixx3g 10234	. . . . 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 1036	. 2 |- ( ( C e. ( A [,] B ) /\ D e. ( A [,] B ) ) -> A e. RR* )
6	4	simp2d 1037	. 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 10234	. . . . 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 10141	. . 3 |- ( ( A e. RR* /\ C e. RR* /\ w e. RR* ) -> ( ( A <_ C /\ C <_ w ) -> A <_ w ) )
15		xrletr 10141	. . 3 |- ( ( w e. RR* /\ D e. RR* /\ B e. RR* ) -> ( ( w <_ D /\ D <_ B ) -> w <_ B ) )
16	1, 1, 14, 15	ixxss12 10239	. 2 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( A <_ C /\ D <_ B ) ) -> ( C [,] D ) C_ ( A [,] B ) )
17	5, 6, 9, 13, 16	syl22anc 1275	1 |- ( ( C e. ( A [,] B ) /\ D e. ( A [,] B ) ) -> ( C [,] D ) C_ ( A [,] B ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   e. wcel 2203   C_ wss 3211   class class class wbr 4109  (class class class)co 6050   RR*cxr 8307   <_ cle 8309   [,]cicc 10224

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-id 4414  df-po 4417  df-iso 4418  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-iota 5312  df-fun 5354  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-icc 10228

This theorem is referenced by: (None)