Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rencldnfi Unicode version

Theorem rencldnfi 26227
Description: A set of real numbers which comes arbitrarily close to some target yet excludes it is infinite. The work is done in rencldnfilem 26226 using infima; this theorem removes the requirement that A be non-empty. (Contributed by Stefan O'Rear, 19-Oct-2014.)
Assertion
Ref Expression
rencldnfi  |-  ( ( ( A  C_  RR  /\  B  e.  RR  /\  -.  B  e.  A
)  /\  A. x  e.  RR+  E. y  e.  A  ( abs `  (
y  -  B ) )  <  x )  ->  -.  A  e.  Fin )
Distinct variable groups:    x, A, y    x, B, y

Proof of Theorem rencldnfi
StepHypRef Expression
1 simpl1 958 . 2  |-  ( ( ( A  C_  RR  /\  B  e.  RR  /\  -.  B  e.  A
)  /\  A. x  e.  RR+  E. y  e.  A  ( abs `  (
y  -  B ) )  <  x )  ->  A  C_  RR )
2 simpl2 959 . 2  |-  ( ( ( A  C_  RR  /\  B  e.  RR  /\  -.  B  e.  A
)  /\  A. x  e.  RR+  E. y  e.  A  ( abs `  (
y  -  B ) )  <  x )  ->  B  e.  RR )
3 rexn0 3632 . . . . . 6  |-  ( E. y  e.  A  ( abs `  ( y  -  B ) )  <  x  ->  A  =/=  (/) )
43ralimi 2694 . . . . 5  |-  ( A. x  e.  RR+  E. y  e.  A  ( abs `  ( y  -  B
) )  <  x  ->  A. x  e.  RR+  A  =/=  (/) )
5 1rp 10447 . . . . . 6  |-  1  e.  RR+
6 ne0i 3537 . . . . . 6  |-  ( 1  e.  RR+  ->  RR+  =/=  (/) )
7 r19.3rzv 3623 . . . . . 6  |-  ( RR+  =/=  (/)  ->  ( A  =/=  (/)  <->  A. x  e.  RR+  A  =/=  (/) ) )
85, 6, 7mp2b 9 . . . . 5  |-  ( A  =/=  (/)  <->  A. x  e.  RR+  A  =/=  (/) )
94, 8sylibr 203 . . . 4  |-  ( A. x  e.  RR+  E. y  e.  A  ( abs `  ( y  -  B
) )  <  x  ->  A  =/=  (/) )
109adantl 452 . . 3  |-  ( ( ( A  C_  RR  /\  B  e.  RR  /\  -.  B  e.  A
)  /\  A. x  e.  RR+  E. y  e.  A  ( abs `  (
y  -  B ) )  <  x )  ->  A  =/=  (/) )
11 simpl3 960 . . 3  |-  ( ( ( A  C_  RR  /\  B  e.  RR  /\  -.  B  e.  A
)  /\  A. x  e.  RR+  E. y  e.  A  ( abs `  (
y  -  B ) )  <  x )  ->  -.  B  e.  A )
1210, 11jca 518 . 2  |-  ( ( ( A  C_  RR  /\  B  e.  RR  /\  -.  B  e.  A
)  /\  A. x  e.  RR+  E. y  e.  A  ( abs `  (
y  -  B ) )  <  x )  ->  ( A  =/=  (/)  /\  -.  B  e.  A ) )
13 simpr 447 . 2  |-  ( ( ( A  C_  RR  /\  B  e.  RR  /\  -.  B  e.  A
)  /\  A. x  e.  RR+  E. y  e.  A  ( abs `  (
y  -  B ) )  <  x )  ->  A. x  e.  RR+  E. y  e.  A  ( abs `  ( y  -  B ) )  <  x )
14 rencldnfilem 26226 . 2  |-  ( ( ( A  C_  RR  /\  B  e.  RR  /\  ( A  =/=  (/)  /\  -.  B  e.  A )
)  /\  A. x  e.  RR+  E. y  e.  A  ( abs `  (
y  -  B ) )  <  x )  ->  -.  A  e.  Fin )
151, 2, 12, 13, 14syl31anc 1185 1  |-  ( ( ( A  C_  RR  /\  B  e.  RR  /\  -.  B  e.  A
)  /\  A. x  e.  RR+  E. y  e.  A  ( abs `  (
y  -  B ) )  <  x )  ->  -.  A  e.  Fin )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    e. wcel 1710    =/= wne 2521   A.wral 2619   E.wrex 2620    C_ wss 3228   (/)c0 3531   class class class wbr 4102   ` cfv 5334  (class class class)co 5942   Fincfn 6948   RRcr 8823   1c1 8825    < clt 8954    - cmin 9124   RR+crp 10443   abscabs 11809
This theorem is referenced by:  irrapx1  26236
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901  ax-pre-sup 8902
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-int 3942  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-riota 6388  df-recs 6472  df-rdg 6507  df-1o 6563  df-oadd 6567  df-er 6744  df-en 6949  df-dom 6950  df-sdom 6951  df-fin 6952  df-sup 7281  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-div 9511  df-nn 9834  df-2 9891  df-3 9892  df-n0 10055  df-z 10114  df-uz 10320  df-rp 10444  df-seq 11136  df-exp 11195  df-cj 11674  df-re 11675  df-im 11676  df-sqr 11810  df-abs 11811
  Copyright terms: Public domain W3C validator