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Theorem negn0 11055
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 3757 . . 3  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
2 ssel 3461 . . . . . . 7  |-  ( A 
C_  RR  ->  ( x  e.  A  ->  x  e.  RR ) )
3 renegcl 9786 . . . . . . . . . 10  |-  ( x  e.  RR  ->  -u x  e.  RR )
4 negeq 9716 . . . . . . . . . . . 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 3225 . . . . . . . . . 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 9486 . . . . . . . . . . 11  |-  ( x  e.  RR  ->  x  e.  CC )
98negnegd 9824 . . . . . . . . . 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 3090 . . . . . 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 3757 . . . . 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 1691 . . 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 429 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 369    = wceq 1370   E.wex 1587    e. wcel 1758    =/= wne 2648   {crab 2803    C_ wss 3439   (/)c0 3748   RRcr 9395   -ucneg 9710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-po 4752  df-so 4753  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-pnf 9534  df-mnf 9535  df-ltxr 9537  df-sub 9711  df-neg 9712
This theorem is referenced by:  supminf  11056
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