Theorem eliooxr 10123

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 10088	. 2 |- (,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x < z /\ z < y ) } )
2	1	elmpocl 6200	1 |- ( A e. ( B (,) C ) -> ( B e. RR* /\ C e. RR* ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2200   {crab 2512   class class class wbr 4083  (class class class)co 6001   RR*cxr 8180   < clt 8181   (,)cioo 10084

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 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-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  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-ral 2513  df-rex 2514  df-v 2801  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-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 6004  df-oprab 6005  df-mpo 6006  df-ioo 10088

This theorem is referenced by:  eliooord  10124  elioo4g  10130  iccssioo2  10142  tgioo  15228