Theorem iocssioo 9977

Description: Condition for a closed interval to be a subset of an open interval. (Contributed by Thierry Arnoux, 29-Mar-2017.)

Assertion

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

Proof of Theorem iocssioo

Dummy variables a b w x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ioo 9906	. 2 |- (,) = ( a e. RR* , b e. RR* |-> { x e. RR* | ( a < x /\ x < b ) } )
2		df-ioc 9907	. 2 |- (,] = ( a e. RR* , b e. RR* |-> { x e. RR* | ( a < x /\ x <_ b ) } )
3		xrlelttr 9820	. 2 |- ( ( A e. RR* /\ C e. RR* /\ w e. RR* ) -> ( ( A <_ C /\ C < w ) -> A < w ) )
4		xrlelttr 9820	. 2 |- ( ( w e. RR* /\ D e. RR* /\ B e. RR* ) -> ( ( w <_ D /\ D < B ) -> w < B ) )
5	1, 2, 3, 4	ixxss12 9920	1 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( A <_ C /\ D < B ) ) -> ( C (,] D ) C_ ( A (,) B ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2158   C_ wss 3141   class class class wbr 4015  (class class class)co 5888   RR*cxr 8005   < clt 8006   <_ cle 8007   (,)cioo 9902   (,]cioc 9903

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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7916  ax-resscn 7917  ax-pre-ltirr 7937  ax-pre-ltwlin 7938  ax-pre-lttrn 7939

This theorem depends on definitions:  df-bi 117  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-sbc 2975  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-id 4305  df-po 4308  df-iso 4309  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-iota 5190  df-fun 5230  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-pnf 8008  df-mnf 8009  df-xr 8010  df-ltxr 8011  df-le 8012  df-ioo 9906  df-ioc 9907

This theorem is referenced by: (None)