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Theorem ioom 9592
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 9252 . . . . . . . 8  |-  ( x  e.  ( A (,) B )  <->  ( ( A  e.  RR*  /\  B  e.  RR*  /\  x  e. 
RR* )  /\  ( A  <  x  /\  x  <  B ) ) )
21biimpi 118 . . . . . . 7  |-  ( x  e.  ( A (,) B )  ->  (
( A  e.  RR*  /\  B  e.  RR*  /\  x  e.  RR* )  /\  ( A  <  x  /\  x  <  B ) ) )
32simpld 110 . . . . . 6  |-  ( x  e.  ( A (,) B )  ->  ( A  e.  RR*  /\  B  e.  RR*  /\  x  e. 
RR* ) )
43simp1d 953 . . . . 5  |-  ( x  e.  ( A (,) B )  ->  A  e.  RR* )
53simp3d 955 . . . . 5  |-  ( x  e.  ( A (,) B )  ->  x  e.  RR* )
63simp2d 954 . . . . 5  |-  ( x  e.  ( A (,) B )  ->  B  e.  RR* )
72simprd 112 . . . . . 6  |-  ( x  e.  ( A (,) B )  ->  ( A  <  x  /\  x  <  B ) )
87simpld 110 . . . . 5  |-  ( x  e.  ( A (,) B )  ->  A  <  x )
97simprd 112 . . . . 5  |-  ( x  e.  ( A (,) B )  ->  x  <  B )
104, 5, 6, 8, 9xrlttrd 9198 . . . 4  |-  ( x  e.  ( A (,) B )  ->  A  <  B )
1110a1i 9 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
x  e.  ( A (,) B )  ->  A  <  B ) )
1211exlimdv 1744 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( E. x  x  e.  ( A (,) B )  ->  A  <  B
) )
13 qbtwnxr 9589 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) )
14 df-rex 2361 . . . . 5  |-  ( E. x  e.  QQ  ( A  <  x  /\  x  <  B )  <->  E. x
( x  e.  QQ  /\  ( A  <  x  /\  x  <  B ) ) )
1513, 14sylib 120 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  E. x
( x  e.  QQ  /\  ( A  <  x  /\  x  <  B ) ) )
16 simpl1 944 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  (
x  e.  QQ  /\  ( A  <  x  /\  x  <  B ) ) )  ->  A  e.  RR* )
17 simpl2 945 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  (
x  e.  QQ  /\  ( A  <  x  /\  x  <  B ) ) )  ->  B  e.  RR* )
18 qre 9034 . . . . . . . . 9  |-  ( x  e.  QQ  ->  x  e.  RR )
1918ad2antrl 474 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  (
x  e.  QQ  /\  ( A  <  x  /\  x  <  B ) ) )  ->  x  e.  RR )
2019rexrd 7473 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  (
x  e.  QQ  /\  ( A  <  x  /\  x  <  B ) ) )  ->  x  e.  RR* )
21 simprrl 506 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  (
x  e.  QQ  /\  ( A  <  x  /\  x  <  B ) ) )  ->  A  <  x )
22 simprrr 507 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  (
x  e.  QQ  /\  ( A  <  x  /\  x  <  B ) ) )  ->  x  <  B )
231biimpri 131 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  x  e.  RR* )  /\  ( A  <  x  /\  x  <  B ) )  ->  x  e.  ( A (,) B ) )
2416, 17, 20, 21, 22, 23syl32anc 1180 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  (
x  e.  QQ  /\  ( A  <  x  /\  x  <  B ) ) )  ->  x  e.  ( A (,) B ) )
2524ex 113 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  (
( x  e.  QQ  /\  ( A  <  x  /\  x  <  B ) )  ->  x  e.  ( A (,) B ) ) )
2625eximdv 1805 . . . 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 1143 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  ->  E. x  x  e.  ( A (,) B ) ) )
2912, 28impbid 127 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 102    <-> wb 103    /\ w3a 922   E.wex 1424    e. wcel 1436   E.wrex 2356   class class class wbr 3819  (class class class)co 5606   RRcr 7285   RR*cxr 7457    < clt 7458   QQcq 9028   (,)cioo 9230
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3930  ax-pow 3982  ax-pr 4008  ax-un 4232  ax-setind 4324  ax-cnex 7372  ax-resscn 7373  ax-1cn 7374  ax-1re 7375  ax-icn 7376  ax-addcl 7377  ax-addrcl 7378  ax-mulcl 7379  ax-mulrcl 7380  ax-addcom 7381  ax-mulcom 7382  ax-addass 7383  ax-mulass 7384  ax-distr 7385  ax-i2m1 7386  ax-0lt1 7387  ax-1rid 7388  ax-0id 7389  ax-rnegex 7390  ax-precex 7391  ax-cnre 7392  ax-pre-ltirr 7393  ax-pre-ltwlin 7394  ax-pre-lttrn 7395  ax-pre-apti 7396  ax-pre-ltadd 7397  ax-pre-mulgt0 7398  ax-pre-mulext 7399  ax-arch 7400
This theorem depends on definitions:  df-bi 115  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-nel 2347  df-ral 2360  df-rex 2361  df-reu 2362  df-rmo 2363  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-pw 3416  df-sn 3436  df-pr 3437  df-op 3439  df-uni 3636  df-int 3671  df-iun 3714  df-br 3820  df-opab 3874  df-mpt 3875  df-id 4092  df-po 4095  df-iso 4096  df-xp 4415  df-rel 4416  df-cnv 4417  df-co 4418  df-dm 4419  df-rn 4420  df-res 4421  df-ima 4422  df-iota 4942  df-fun 4979  df-fn 4980  df-f 4981  df-fv 4985  df-riota 5562  df-ov 5609  df-oprab 5610  df-mpt2 5611  df-1st 5861  df-2nd 5862  df-pnf 7460  df-mnf 7461  df-xr 7462  df-ltxr 7463  df-le 7464  df-sub 7591  df-neg 7592  df-reap 7985  df-ap 7992  df-div 8071  df-inn 8350  df-2 8408  df-n0 8599  df-z 8676  df-uz 8944  df-q 9029  df-rp 9059  df-ioo 9234
This theorem is referenced by: (None)
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