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Theorem negn0 9999
Description: The image under negation of a nonempty set of reals is nonempty. (Contributed by Paul Chapman, 21-Mar-2011.)
Assertion
Ref Expression
negn0  |-  ( ( A  C_  RR  /\  A  =/=  (/) )  ->  { z  e.  RR  |  -u z  e.  A }  =/=  (/) )
Distinct variable group:    z, A

Proof of Theorem negn0
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 n0 3714 . . 3  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
2 ssel 3401 . . . . . . 7  |-  ( A 
C_  RR  ->  ( x  e.  A  ->  x  e.  RR ) )
3 renegcl 9888 . . . . . . . . . 10  |-  ( x  e.  RR  ->  -u x  e.  RR )
4 negeq 9818 . . . . . . . . . . . 12  |-  ( z  =  -u x  ->  -u z  =  -u -u x )
54eleq1d 2490 . . . . . . . . . . 11  |-  ( z  =  -u x  ->  ( -u z  e.  A  <->  -u -u x  e.  A ) )
65elrab3 3172 . . . . . . . . . 10  |-  ( -u x  e.  RR  ->  (
-u x  e.  {
z  e.  RR  |  -u z  e.  A }  <->  -u -u x  e.  A
) )
73, 6syl 17 . . . . . . . . 9  |-  ( x  e.  RR  ->  ( -u x  e.  { z  e.  RR  |  -u z  e.  A }  <->  -u -u x  e.  A
) )
8 recn 9580 . . . . . . . . . . 11  |-  ( x  e.  RR  ->  x  e.  CC )
98negnegd 9928 . . . . . . . . . 10  |-  ( x  e.  RR  ->  -u -u x  =  x )
109eleq1d 2490 . . . . . . . . 9  |-  ( x  e.  RR  ->  ( -u -u x  e.  A  <->  x  e.  A ) )
117, 10bitrd 256 . . . . . . . 8  |-  ( x  e.  RR  ->  ( -u x  e.  { z  e.  RR  |  -u z  e.  A }  <->  x  e.  A ) )
1211biimprd 226 . . . . . . 7  |-  ( x  e.  RR  ->  (
x  e.  A  ->  -u x  e.  { z  e.  RR  |  -u z  e.  A }
) )
132, 12syli 38 . . . . . 6  |-  ( A 
C_  RR  ->  ( x  e.  A  ->  -u x  e.  { z  e.  RR  |  -u z  e.  A } ) )
14 elex2 3034 . . . . . 6  |-  ( -u x  e.  { z  e.  RR  |  -u z  e.  A }  ->  E. y 
y  e.  { z  e.  RR  |  -u z  e.  A }
)
1513, 14syl6 34 . . . . 5  |-  ( A 
C_  RR  ->  ( x  e.  A  ->  E. y 
y  e.  { z  e.  RR  |  -u z  e.  A }
) )
16 n0 3714 . . . . 5  |-  ( { z  e.  RR  |  -u z  e.  A }  =/=  (/)  <->  E. y  y  e. 
{ z  e.  RR  |  -u z  e.  A } )
1715, 16syl6ibr 230 . . . 4  |-  ( A 
C_  RR  ->  ( x  e.  A  ->  { z  e.  RR  |  -u z  e.  A }  =/=  (/) ) )
1817exlimdv 1772 . . 3  |-  ( A 
C_  RR  ->  ( E. x  x  e.  A  ->  { z  e.  RR  |  -u z  e.  A }  =/=  (/) ) )
191, 18syl5bi 220 . 2  |-  ( A 
C_  RR  ->  ( A  =/=  (/)  ->  { z  e.  RR  |  -u z  e.  A }  =/=  (/) ) )
2019imp 430 1  |-  ( ( A  C_  RR  /\  A  =/=  (/) )  ->  { z  e.  RR  |  -u z  e.  A }  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437   E.wex 1657    e. wcel 1872    =/= wne 2599   {crab 2718    C_ wss 3379   (/)c0 3704   RRcr 9489   -ucneg 9812
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-nel 2602  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-br 4367  df-opab 4426  df-mpt 4427  df-id 4711  df-po 4717  df-so 4718  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-riota 6211  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-er 7318  df-en 7525  df-dom 7526  df-sdom 7527  df-pnf 9628  df-mnf 9629  df-ltxr 9631  df-sub 9813  df-neg 9814
This theorem is referenced by:  fiminre  10506  supminf  11201  supminfOLD  11202
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