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Theorem nsmallnq 9344
Description: The is no smallest positive fraction. (Contributed by NM, 26-Apr-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
nsmallnq  |-  ( A  e.  Q.  ->  E. x  x  <Q  A )
Distinct variable group:    x, A

Proof of Theorem nsmallnq
StepHypRef Expression
1 halfnq 9343 . 2  |-  ( A  e.  Q.  ->  E. x
( x  +Q  x
)  =  A )
2 eleq1a 2537 . . . . 5  |-  ( A  e.  Q.  ->  (
( x  +Q  x
)  =  A  -> 
( x  +Q  x
)  e.  Q. )
)
3 addnqf 9315 . . . . . . . 8  |-  +Q  :
( Q.  X.  Q. )
--> Q.
43fdmi 5718 . . . . . . 7  |-  dom  +Q  =  ( Q.  X.  Q. )
5 0nnq 9291 . . . . . . 7  |-  -.  (/)  e.  Q.
64, 5ndmovrcl 6434 . . . . . 6  |-  ( ( x  +Q  x )  e.  Q.  ->  (
x  e.  Q.  /\  x  e.  Q. )
)
7 ltaddnq 9341 . . . . . 6  |-  ( ( x  e.  Q.  /\  x  e.  Q. )  ->  x  <Q  ( x  +Q  x ) )
86, 7syl 16 . . . . 5  |-  ( ( x  +Q  x )  e.  Q.  ->  x  <Q  ( x  +Q  x
) )
92, 8syl6 33 . . . 4  |-  ( A  e.  Q.  ->  (
( x  +Q  x
)  =  A  ->  x  <Q  ( x  +Q  x ) ) )
10 breq2 4443 . . . 4  |-  ( ( x  +Q  x )  =  A  ->  (
x  <Q  ( x  +Q  x )  <->  x  <Q  A ) )
119, 10mpbidi 216 . . 3  |-  ( A  e.  Q.  ->  (
( x  +Q  x
)  =  A  ->  x  <Q  A ) )
1211eximdv 1715 . 2  |-  ( A  e.  Q.  ->  ( E. x ( x  +Q  x )  =  A  ->  E. x  x  <Q  A ) )
131, 12mpd 15 1  |-  ( A  e.  Q.  ->  E. x  x  <Q  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398   E.wex 1617    e. wcel 1823   class class class wbr 4439    X. cxp 4986  (class class class)co 6270   Q.cnq 9219    +Q cplq 9222    <Q cltq 9225
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-omul 7127  df-er 7303  df-ni 9239  df-pli 9240  df-mi 9241  df-lti 9242  df-plpq 9275  df-mpq 9276  df-ltpq 9277  df-enq 9278  df-nq 9279  df-erq 9280  df-plq 9281  df-mq 9282  df-1nq 9283  df-rq 9284  df-ltnq 9285
This theorem is referenced by:  ltbtwnnq  9345  nqpr  9381  reclem2pr  9415
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