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Theorem ndmioo 10836
Description: The open interval function's value is empty outside of its domain. (Contributed by NM, 21-Jun-2007.) (Revised by Mario Carneiro, 28-Apr-2015.)
Assertion
Ref Expression
ndmioo  |-  ( -.  ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A (,) B
)  =  (/) )

Proof of Theorem ndmioo
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ioo 10813 . . . 4  |-  (,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <  y ) } )
21ixxf 10819 . . 3  |-  (,) :
( RR*  X.  RR* ) --> ~P RR*
32fdmi 5500 . 2  |-  dom  (,)  =  ( RR*  X.  RR* )
43ndmov 6131 1  |-  ( -.  ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A (,) B
)  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1647    e. wcel 1715   (/)c0 3543   ~Pcpw 3714    X. cxp 4790  (class class class)co 5981   RR*cxr 9013    < clt 9014   (,)cioo 10809
This theorem is referenced by:  iooid  10837  eliooxr  10862  iccssioo2  10875  ioombl  19137  mbfima  19202  dvferm1lem  19546  dvferm2lem  19548  dvferm  19550  dvivthlem1  19570
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-cnex 8940  ax-resscn 8941
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-xr 9018  df-ioo 10813
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