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Theorem negn0 11169
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 3793 . . 3  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
2 ssel 3483 . . . . . . 7  |-  ( A 
C_  RR  ->  ( x  e.  A  ->  x  e.  RR ) )
3 renegcl 9873 . . . . . . . . . 10  |-  ( x  e.  RR  ->  -u x  e.  RR )
4 negeq 9803 . . . . . . . . . . . 12  |-  ( z  =  -u x  ->  -u z  =  -u -u x )
54eleq1d 2523 . . . . . . . . . . 11  |-  ( z  =  -u x  ->  ( -u z  e.  A  <->  -u -u x  e.  A ) )
65elrab3 3255 . . . . . . . . . 10  |-  ( -u x  e.  RR  ->  (
-u x  e.  {
z  e.  RR  |  -u z  e.  A }  <->  -u -u x  e.  A
) )
73, 6syl 16 . . . . . . . . 9  |-  ( x  e.  RR  ->  ( -u x  e.  { z  e.  RR  |  -u z  e.  A }  <->  -u -u x  e.  A
) )
8 recn 9571 . . . . . . . . . . 11  |-  ( x  e.  RR  ->  x  e.  CC )
98negnegd 9913 . . . . . . . . . 10  |-  ( x  e.  RR  ->  -u -u x  =  x )
109eleq1d 2523 . . . . . . . . 9  |-  ( x  e.  RR  ->  ( -u -u x  e.  A  <->  x  e.  A ) )
117, 10bitrd 253 . . . . . . . 8  |-  ( x  e.  RR  ->  ( -u x  e.  { z  e.  RR  |  -u z  e.  A }  <->  x  e.  A ) )
1211biimprd 223 . . . . . . 7  |-  ( x  e.  RR  ->  (
x  e.  A  ->  -u x  e.  { z  e.  RR  |  -u z  e.  A }
) )
132, 12syli 37 . . . . . 6  |-  ( A 
C_  RR  ->  ( x  e.  A  ->  -u x  e.  { z  e.  RR  |  -u z  e.  A } ) )
14 elex2 3118 . . . . . 6  |-  ( -u x  e.  { z  e.  RR  |  -u z  e.  A }  ->  E. y 
y  e.  { z  e.  RR  |  -u z  e.  A }
)
1513, 14syl6 33 . . . . 5  |-  ( A 
C_  RR  ->  ( x  e.  A  ->  E. y 
y  e.  { z  e.  RR  |  -u z  e.  A }
) )
16 n0 3793 . . . . 5  |-  ( { z  e.  RR  |  -u z  e.  A }  =/=  (/)  <->  E. y  y  e. 
{ z  e.  RR  |  -u z  e.  A } )
1715, 16syl6ibr 227 . . . 4  |-  ( A 
C_  RR  ->  ( x  e.  A  ->  { z  e.  RR  |  -u z  e.  A }  =/=  (/) ) )
1817exlimdv 1729 . . 3  |-  ( A 
C_  RR  ->  ( E. x  x  e.  A  ->  { z  e.  RR  |  -u z  e.  A }  =/=  (/) ) )
191, 18syl5bi 217 . 2  |-  ( A 
C_  RR  ->  ( A  =/=  (/)  ->  { z  e.  RR  |  -u z  e.  A }  =/=  (/) ) )
2019imp 427 1  |-  ( ( A  C_  RR  /\  A  =/=  (/) )  ->  { z  e.  RR  |  -u z  e.  A }  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398   E.wex 1617    e. wcel 1823    =/= wne 2649   {crab 2808    C_ wss 3461   (/)c0 3783   RRcr 9480   -ucneg 9797
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  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557
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-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  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-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-po 4789  df-so 4790  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-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-ltxr 9622  df-sub 9798  df-neg 9799
This theorem is referenced by:  supminf  11170
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