Theorem iccssico2 10155

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 10102	. . . 4 |- [,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x <_ z /\ z < y ) } )
2	1	elmpocl1 6207	. . 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 6208	. . 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 10109	. . . . 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 10109	. . . . 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 10153	. 2 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( A <_ C /\ D < B ) ) -> ( C [,] D ) C_ ( A [,) B ) )
15	3, 5, 9, 13, 14	syl22anc 1272	1 |- ( ( C e. ( A [,) B ) /\ D e. ( A [,) B ) ) -> ( C [,] D ) C_ ( A [,) B ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002   e. wcel 2200   {crab 2512   C_ wss 3197   class class class wbr 4083  (class class class)co 6007   RR*cxr 8191   < clt 8192   <_ cle 8193   [,)cico 10098   [,]cicc 10099

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8101  ax-resscn 8102  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-id 4384  df-po 4387  df-iso 4388  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fv 5326  df-ov 6010  df-oprab 6011  df-mpo 6012  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-ico 10102  df-icc 10103

This theorem is referenced by: (None)