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Theorem ioom 9931
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 9586 . . . . . . . 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 976 . . . . 5  |-  ( x  e.  ( A (,) B )  ->  A  e.  RR* )
53simp3d 978 . . . . 5  |-  ( x  e.  ( A (,) B )  ->  x  e.  RR* )
63simp2d 977 . . . . 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 9485 . . . 4  |-  ( x  e.  ( A (,) B )  ->  A  <  B )
1110a1i 9 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
x  e.  ( A (,) B )  ->  A  <  B ) )
1211exlimdv 1773 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( E. x  x  e.  ( A (,) B )  ->  A  <  B
) )
13 qbtwnxr 9928 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) )
14 df-rex 2396 . . . . 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 967 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  (
x  e.  QQ  /\  ( A  <  x  /\  x  <  B ) ) )  ->  A  e.  RR* )
17 simpl2 968 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  (
x  e.  QQ  /\  ( A  <  x  /\  x  <  B ) ) )  ->  B  e.  RR* )
18 qre 9319 . . . . . . . . 9  |-  ( x  e.  QQ  ->  x  e.  RR )
1918ad2antrl 479 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  (
x  e.  QQ  /\  ( A  <  x  /\  x  <  B ) ) )  ->  x  e.  RR )
2019rexrd 7739 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  (
x  e.  QQ  /\  ( A  <  x  /\  x  <  B ) ) )  ->  x  e.  RR* )
21 simprrl 511 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  (
x  e.  QQ  /\  ( A  <  x  /\  x  <  B ) ) )  ->  A  <  x )
22 simprrr 512 . . . . . . 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 1207 . . . . . 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 1834 . . . 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 1166 . 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 945   E.wex 1451    e. wcel 1463   E.wrex 2391   class class class wbr 3895  (class class class)co 5728   RRcr 7546   RR*cxr 7723    < clt 7724   QQcq 9313   (,)cioo 9564
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-cnex 7636  ax-resscn 7637  ax-1cn 7638  ax-1re 7639  ax-icn 7640  ax-addcl 7641  ax-addrcl 7642  ax-mulcl 7643  ax-mulrcl 7644  ax-addcom 7645  ax-mulcom 7646  ax-addass 7647  ax-mulass 7648  ax-distr 7649  ax-i2m1 7650  ax-0lt1 7651  ax-1rid 7652  ax-0id 7653  ax-rnegex 7654  ax-precex 7655  ax-cnre 7656  ax-pre-ltirr 7657  ax-pre-ltwlin 7658  ax-pre-lttrn 7659  ax-pre-apti 7660  ax-pre-ltadd 7661  ax-pre-mulgt0 7662  ax-pre-mulext 7663  ax-arch 7664
This theorem depends on definitions:  df-bi 116  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-nel 2378  df-ral 2395  df-rex 2396  df-reu 2397  df-rmo 2398  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-id 4175  df-po 4178  df-iso 4179  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-fv 5089  df-riota 5684  df-ov 5731  df-oprab 5732  df-mpo 5733  df-1st 5992  df-2nd 5993  df-pnf 7726  df-mnf 7727  df-xr 7728  df-ltxr 7729  df-le 7730  df-sub 7858  df-neg 7859  df-reap 8255  df-ap 8262  df-div 8346  df-inn 8631  df-2 8689  df-n0 8882  df-z 8959  df-uz 9229  df-q 9314  df-rp 9344  df-ioo 9568
This theorem is referenced by:  tgioo  12532
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