Theorem eliooxr 9925

Description: An inhabited open interval spans an interval of extended reals. (Contributed by NM, 17-Aug-2008.)

Assertion

Ref	Expression
eliooxr	|- ( A e. ( B (,) C ) -> ( B e. RR* /\ C e. RR* ) )

Proof of Theorem eliooxr

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

Step	Hyp	Ref	Expression
1		df-ioo 9890	. 2 |- (,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x < z /\ z < y ) } )
2	1	elmpocl 6068	1 |- ( A e. ( B (,) C ) -> ( B e. RR* /\ C e. RR* ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2148   {crab 2459   class class class wbr 4003  (class class class)co 5874   RR*cxr 7989   < clt 7990   (,)cioo 9886

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-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-iota 5178  df-fun 5218  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-ioo 9890

This theorem is referenced by:  eliooord  9926  elioo4g  9932  iccssioo2  9944  tgioo  13977