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Theorem iooidg 9883
Description: An open interval with identical lower and upper bounds is empty. (Contributed by Jim Kingdon, 29-Mar-2020.)
Assertion
Ref Expression
iooidg  |-  ( A  e.  RR*  ->  ( A (,) A )  =  (/) )

Proof of Theorem iooidg
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 iooval 9882 . . 3  |-  ( ( A  e.  RR*  /\  A  e.  RR* )  ->  ( A (,) A )  =  { x  e.  RR*  |  ( A  <  x  /\  x  <  A ) } )
21anidms 397 . 2  |-  ( A  e.  RR*  ->  ( A (,) A )  =  { x  e.  RR*  |  ( A  <  x  /\  x  <  A ) } )
3 xrltnsym2 9768 . . . 4  |-  ( ( A  e.  RR*  /\  x  e.  RR* )  ->  -.  ( A  <  x  /\  x  <  A ) )
43ralrimiva 2550 . . 3  |-  ( A  e.  RR*  ->  A. x  e.  RR*  -.  ( A  <  x  /\  x  <  A ) )
5 rabeq0 3452 . . 3  |-  ( { x  e.  RR*  |  ( A  <  x  /\  x  <  A ) }  =  (/)  <->  A. x  e.  RR*  -.  ( A  <  x  /\  x  <  A ) )
64, 5sylibr 134 . 2  |-  ( A  e.  RR*  ->  { x  e.  RR*  |  ( A  <  x  /\  x  <  A ) }  =  (/) )
72, 6eqtrd 2210 1  |-  ( A  e.  RR*  ->  ( A (,) A )  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148   A.wral 2455   {crab 2459   (/)c0 3422   class class class wbr 4000  (class class class)co 5868   RR*cxr 7968    < clt 7969   (,)cioo 9862
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-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-cnex 7880  ax-resscn 7881  ax-pre-ltirr 7901  ax-pre-lttrn 7903
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-iota 5173  df-fun 5213  df-fv 5219  df-ov 5871  df-oprab 5872  df-mpo 5873  df-pnf 7971  df-mnf 7972  df-xr 7973  df-ltxr 7974  df-ioo 9866
This theorem is referenced by:  blssioo  13678
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