Theorem iccss 10166

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, 20-Feb-2015.)

Assertion

Ref	Expression
iccss	|- ( ( ( A e. RR /\ B e. RR ) /\ ( A <_ C /\ D <_ B ) ) -> ( C [,] D ) C_ ( A [,] B ) )

Proof of Theorem iccss

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

Step	Hyp	Ref	Expression
1		rexr 8215	. . 3 |- ( A e. RR -> A e. RR* )
2		rexr 8215	. . 3 |- ( B e. RR -> B e. RR* )
3	1, 2	anim12i 338	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( A e. RR* /\ B e. RR* ) )
4		df-icc 10120	. . 3 |- [,] = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x <_ z /\ z <_ y ) } )
5		xrletr 10033	. . 3 |- ( ( A e. RR* /\ C e. RR* /\ w e. RR* ) -> ( ( A <_ C /\ C <_ w ) -> A <_ w ) )
6		xrletr 10033	. . 3 |- ( ( w e. RR* /\ D e. RR* /\ B e. RR* ) -> ( ( w <_ D /\ D <_ B ) -> w <_ B ) )
7	4, 4, 5, 6	ixxss12 10131	. 2 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( A <_ C /\ D <_ B ) ) -> ( C [,] D ) C_ ( A [,] B ) )
8	3, 7	sylan 283	1 |- ( ( ( A e. RR /\ B e. RR ) /\ ( A <_ C /\ D <_ B ) ) -> ( C [,] D ) C_ ( A [,] B ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2200   C_ wss 3198   class class class wbr 4086  (class class class)co 6013   RRcr 8021   RR*cxr 8203   <_ cle 8205   [,]cicc 10116

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 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136

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 2802  df-sbc 3030  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-id 4388  df-po 4391  df-iso 4392  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-iota 5284  df-fun 5326  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-icc 10120

This theorem is referenced by: (None)