Theorem iccssico2 10299

Description: Condition for a closed interval to be a subset of a closed-below, open-above interval. (Contributed by Mario Carneiro, 20-Feb-2015.)

Assertion

Ref	Expression
iccssico2	|- ( ( C e. ( A [,) B ) /\ D e. ( A [,) B ) ) -> ( C [,] D ) C_ ( A [,) B ) )

Proof of Theorem iccssico2

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

Step	Hyp	Ref	Expression
1		df-ico 10246	. . . 4 |- [,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x <_ z /\ z < y ) } )
2	1	elmpocl1 6258	. . 3 |- ( C e. ( A [,) B ) -> A e. RR* )
3	2	adantr 276	. 2 |- ( ( C e. ( A [,) B ) /\ D e. ( A [,) B ) ) -> A e. RR* )
4	1	elmpocl2 6259	. . 3 |- ( C e. ( A [,) B ) -> B e. RR* )
5	4	adantr 276	. 2 |- ( ( C e. ( A [,) B ) /\ D e. ( A [,) B ) ) -> B e. RR* )
6	1	elixx3g 10253	. . . . 5 |- ( C e. ( A [,) B ) <-> ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A <_ C /\ C < B ) ) )
7	6	simprbi 275	. . . 4 |- ( C e. ( A [,) B ) -> ( A <_ C /\ C < B ) )
8	7	simpld 112	. . 3 |- ( C e. ( A [,) B ) -> A <_ C )
9	8	adantr 276	. 2 |- ( ( C e. ( A [,) B ) /\ D e. ( A [,) B ) ) -> A <_ C )
10	1	elixx3g 10253	. . . . 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		iccssico 10297	. 2 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( A <_ C /\ D < B ) ) -> ( C [,] D ) C_ ( A [,) B ) )
15	3, 5, 9, 13, 14	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 2205   {crab 2526   C_ wss 3214   class class class wbr 4114  (class class class)co 6058   RR*cxr 8323   < clt 8324   <_ cle 8325   [,)cico 10242   [,]cicc 10243

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-id 4419  df-po 4422  df-iso 4423  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-iota 5317  df-fun 5359  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-ico 10246  df-icc 10247

This theorem is referenced by: (None)