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Theorem ioodisjOLD 25554
Description: If the upper bound of one open interval is less than or equal to the lower bound of the other, the intervals are disjoint. (Moved into main set.mm as ioodisj 10854 and may be deleted by mathbox owner, JGH. --NM 29-May-2014.) (Contributed by Jeff Hankins, 13-Jul-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ioodisjOLD  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* ) )  /\  B  <_  C )  -> 
( ( A (,) B )  i^i  ( C (,) D ) )  =  (/) )

Proof of Theorem ioodisjOLD
StepHypRef Expression
1 ioodisj 10854 1  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* ) )  /\  B  <_  C )  -> 
( ( A (,) B )  i^i  ( C (,) D ) )  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710    i^i cin 3227   (/)c0 3531   class class class wbr 4102  (class class class)co 5942   RR*cxr 8953    <_ cle 8955   (,)cioo 10745
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-cnex 8880  ax-resscn 8881  ax-pre-lttri 8898  ax-pre-lttrn 8899
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-po 4393  df-so 4394  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-er 6744  df-en 6949  df-dom 6950  df-sdom 6951  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-ioo 10749  df-icc 10752
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