Theorem eliooxr 10263

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 10228	. 2 |- (,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x < z /\ z < y ) } )
2	1	elmpocl 6251	1 |- ( A e. ( B (,) C ) -> ( B e. RR* /\ C e. RR* ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2205   {crab 2526   class class class wbr 4111  (class class class)co 6052   RR*cxr 8309   < clt 8310   (,)cioo 10224

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-iota 5314  df-fun 5356  df-fv 5362  df-ov 6055  df-oprab 6056  df-mpo 6057  df-ioo 10228

This theorem is referenced by:  eliooord  10264  elioo4g  10270  iccssioo2  10282  tgioo  15436