Theorem eliooxr 10161

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 10126	. 2 |- (,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x < z /\ z < y ) } )
2	1	elmpocl 6216	1 |- ( A e. ( B (,) C ) -> ( B e. RR* /\ C e. RR* ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2202   {crab 2514   class class class wbr 4088  (class class class)co 6017   RR*cxr 8212   < clt 8213   (,)cioo 10122

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-ioo 10126

This theorem is referenced by:  eliooord  10162  elioo4g  10168  iccssioo2  10180  tgioo  15277