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Theorem ioom 10050
Description: An open interval of extended reals is inhabited iff the lower argument is less than the upper argument. (Contributed by Jim Kingdon, 27-Nov-2021.)
Assertion
Ref Expression
ioom  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( E. x  x  e.  ( A (,) B )  <-> 
A  <  B )
)
Distinct variable groups:    x, A    x, B

Proof of Theorem ioom
StepHypRef Expression
1 elioo3g 9705 . . . . . . . 8  |-  ( x  e.  ( A (,) B )  <->  ( ( A  e.  RR*  /\  B  e.  RR*  /\  x  e. 
RR* )  /\  ( A  <  x  /\  x  <  B ) ) )
21biimpi 119 . . . . . . 7  |-  ( x  e.  ( A (,) B )  ->  (
( A  e.  RR*  /\  B  e.  RR*  /\  x  e.  RR* )  /\  ( A  <  x  /\  x  <  B ) ) )
32simpld 111 . . . . . 6  |-  ( x  e.  ( A (,) B )  ->  ( A  e.  RR*  /\  B  e.  RR*  /\  x  e. 
RR* ) )
43simp1d 993 . . . . 5  |-  ( x  e.  ( A (,) B )  ->  A  e.  RR* )
53simp3d 995 . . . . 5  |-  ( x  e.  ( A (,) B )  ->  x  e.  RR* )
63simp2d 994 . . . . 5  |-  ( x  e.  ( A (,) B )  ->  B  e.  RR* )
72simprd 113 . . . . . 6  |-  ( x  e.  ( A (,) B )  ->  ( A  <  x  /\  x  <  B ) )
87simpld 111 . . . . 5  |-  ( x  e.  ( A (,) B )  ->  A  <  x )
97simprd 113 . . . . 5  |-  ( x  e.  ( A (,) B )  ->  x  <  B )
104, 5, 6, 8, 9xrlttrd 9604 . . . 4  |-  ( x  e.  ( A (,) B )  ->  A  <  B )
1110a1i 9 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
x  e.  ( A (,) B )  ->  A  <  B ) )
1211exlimdv 1791 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( E. x  x  e.  ( A (,) B )  ->  A  <  B
) )
13 qbtwnxr 10047 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) )
14 df-rex 2422 . . . . 5  |-  ( E. x  e.  QQ  ( A  <  x  /\  x  <  B )  <->  E. x
( x  e.  QQ  /\  ( A  <  x  /\  x  <  B ) ) )
1513, 14sylib 121 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  E. x
( x  e.  QQ  /\  ( A  <  x  /\  x  <  B ) ) )
16 simpl1 984 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  (
x  e.  QQ  /\  ( A  <  x  /\  x  <  B ) ) )  ->  A  e.  RR* )
17 simpl2 985 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  (
x  e.  QQ  /\  ( A  <  x  /\  x  <  B ) ) )  ->  B  e.  RR* )
18 qre 9429 . . . . . . . . 9  |-  ( x  e.  QQ  ->  x  e.  RR )
1918ad2antrl 481 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  (
x  e.  QQ  /\  ( A  <  x  /\  x  <  B ) ) )  ->  x  e.  RR )
2019rexrd 7827 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  (
x  e.  QQ  /\  ( A  <  x  /\  x  <  B ) ) )  ->  x  e.  RR* )
21 simprrl 528 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  (
x  e.  QQ  /\  ( A  <  x  /\  x  <  B ) ) )  ->  A  <  x )
22 simprrr 529 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  (
x  e.  QQ  /\  ( A  <  x  /\  x  <  B ) ) )  ->  x  <  B )
231biimpri 132 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  x  e.  RR* )  /\  ( A  <  x  /\  x  <  B ) )  ->  x  e.  ( A (,) B ) )
2416, 17, 20, 21, 22, 23syl32anc 1224 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  (
x  e.  QQ  /\  ( A  <  x  /\  x  <  B ) ) )  ->  x  e.  ( A (,) B ) )
2524ex 114 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  (
( x  e.  QQ  /\  ( A  <  x  /\  x  <  B ) )  ->  x  e.  ( A (,) B ) ) )
2625eximdv 1852 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  ( E. x ( x  e.  QQ  /\  ( A  <  x  /\  x  <  B ) )  ->  E. x  x  e.  ( A (,) B ) ) )
2715, 26mpd 13 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  E. x  x  e.  ( A (,) B ) )
28273expia 1183 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  ->  E. x  x  e.  ( A (,) B ) ) )
2912, 28impbid 128 1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( E. x  x  e.  ( A (,) B )  <-> 
A  <  B )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 962   E.wex 1468    e. wcel 1480   E.wrex 2417   class class class wbr 3929  (class class class)co 5774   RRcr 7631   RR*cxr 7811    < clt 7812   QQcq 9423   (,)cioo 9683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7723  ax-resscn 7724  ax-1cn 7725  ax-1re 7726  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-mulrcl 7731  ax-addcom 7732  ax-mulcom 7733  ax-addass 7734  ax-mulass 7735  ax-distr 7736  ax-i2m1 7737  ax-0lt1 7738  ax-1rid 7739  ax-0id 7740  ax-rnegex 7741  ax-precex 7742  ax-cnre 7743  ax-pre-ltirr 7744  ax-pre-ltwlin 7745  ax-pre-lttrn 7746  ax-pre-apti 7747  ax-pre-ltadd 7748  ax-pre-mulgt0 7749  ax-pre-mulext 7750  ax-arch 7751
This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-po 4218  df-iso 4219  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-pnf 7814  df-mnf 7815  df-xr 7816  df-ltxr 7817  df-le 7818  df-sub 7947  df-neg 7948  df-reap 8349  df-ap 8356  df-div 8445  df-inn 8733  df-2 8791  df-n0 8990  df-z 9067  df-uz 9339  df-q 9424  df-rp 9454  df-ioo 9687
This theorem is referenced by:  tgioo  12729
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