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Theorem nsmallnq 8780
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 8779 . 2  |-  ( A  e.  Q.  ->  E. x
( x  +Q  x
)  =  A )
2 eleq1a 2449 . . . . 5  |-  ( A  e.  Q.  ->  (
( x  +Q  x
)  =  A  -> 
( x  +Q  x
)  e.  Q. )
)
3 addnqf 8751 . . . . . . . 8  |-  +Q  :
( Q.  X.  Q. )
--> Q.
43fdmi 5529 . . . . . . 7  |-  dom  +Q  =  ( Q.  X.  Q. )
5 0nnq 8727 . . . . . . 7  |-  -.  (/)  e.  Q.
64, 5ndmovrcl 6165 . . . . . 6  |-  ( ( x  +Q  x )  e.  Q.  ->  (
x  e.  Q.  /\  x  e.  Q. )
)
7 ltaddnq 8777 . . . . . 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 31 . . . 4  |-  ( A  e.  Q.  ->  (
( x  +Q  x
)  =  A  ->  x  <Q  ( x  +Q  x ) ) )
10 breq2 4150 . . . 4  |-  ( ( x  +Q  x )  =  A  ->  (
x  <Q  ( x  +Q  x )  <->  x  <Q  A ) )
119, 10mpbidi 208 . . 3  |-  ( A  e.  Q.  ->  (
( x  +Q  x
)  =  A  ->  x  <Q  A ) )
1211eximdv 1629 . 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 set class
Syntax hints:    -> wi 4    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1717   class class class wbr 4146    X. cxp 4809  (class class class)co 6013   Q.cnq 8653    +Q cplq 8656    <Q cltq 8659
This theorem is referenced by:  ltbtwnnq  8781  nqpr  8817  reclem2pr  8851
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-recs 6562  df-rdg 6597  df-1o 6653  df-oadd 6657  df-omul 6658  df-er 6834  df-ni 8675  df-pli 8676  df-mi 8677  df-lti 8678  df-plpq 8711  df-mpq 8712  df-ltpq 8713  df-enq 8714  df-nq 8715  df-erq 8716  df-plq 8717  df-mq 8718  df-1nq 8719  df-rq 8720  df-ltnq 8721
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