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Theorem nsmallnq 8843
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 8842 . 2  |-  ( A  e.  Q.  ->  E. x
( x  +Q  x
)  =  A )
2 eleq1a 2504 . . . . 5  |-  ( A  e.  Q.  ->  (
( x  +Q  x
)  =  A  -> 
( x  +Q  x
)  e.  Q. )
)
3 addnqf 8814 . . . . . . . 8  |-  +Q  :
( Q.  X.  Q. )
--> Q.
43fdmi 5587 . . . . . . 7  |-  dom  +Q  =  ( Q.  X.  Q. )
5 0nnq 8790 . . . . . . 7  |-  -.  (/)  e.  Q.
64, 5ndmovrcl 6224 . . . . . 6  |-  ( ( x  +Q  x )  e.  Q.  ->  (
x  e.  Q.  /\  x  e.  Q. )
)
7 ltaddnq 8840 . . . . . 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 4208 . . . 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 1632 . 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 1550    = wceq 1652    e. wcel 1725   class class class wbr 4204    X. cxp 4867  (class class class)co 6072   Q.cnq 8716    +Q cplq 8719    <Q cltq 8722
This theorem is referenced by:  ltbtwnnq  8844  nqpr  8880  reclem2pr  8914
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-recs 6624  df-rdg 6659  df-1o 6715  df-oadd 6719  df-omul 6720  df-er 6896  df-ni 8738  df-pli 8739  df-mi 8740  df-lti 8741  df-plpq 8774  df-mpq 8775  df-ltpq 8776  df-enq 8777  df-nq 8778  df-erq 8779  df-plq 8780  df-mq 8781  df-1nq 8782  df-rq 8783  df-ltnq 8784
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