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Theorem negm 9375
Description: The image under negation of an inhabited set of reals is inhabited. (Contributed by Jim Kingdon, 10-Apr-2020.)
Assertion
Ref Expression
negm  |-  ( ( A  C_  RR  /\  E. x  x  e.  A
)  ->  E. y 
y  e.  { z  e.  RR  |  -u z  e.  A }
)
Distinct variable group:    x, A, y, z

Proof of Theorem negm
StepHypRef Expression
1 ssel 3061 . . . . 5  |-  ( A 
C_  RR  ->  ( x  e.  A  ->  x  e.  RR ) )
2 renegcl 7991 . . . . . . . 8  |-  ( x  e.  RR  ->  -u x  e.  RR )
3 negeq 7923 . . . . . . . . . 10  |-  ( z  =  -u x  ->  -u z  =  -u -u x )
43eleq1d 2186 . . . . . . . . 9  |-  ( z  =  -u x  ->  ( -u z  e.  A  <->  -u -u x  e.  A ) )
54elrab3 2814 . . . . . . . 8  |-  ( -u x  e.  RR  ->  (
-u x  e.  {
z  e.  RR  |  -u z  e.  A }  <->  -u -u x  e.  A
) )
62, 5syl 14 . . . . . . 7  |-  ( x  e.  RR  ->  ( -u x  e.  { z  e.  RR  |  -u z  e.  A }  <->  -u -u x  e.  A
) )
7 recn 7721 . . . . . . . . 9  |-  ( x  e.  RR  ->  x  e.  CC )
87negnegd 8032 . . . . . . . 8  |-  ( x  e.  RR  ->  -u -u x  =  x )
98eleq1d 2186 . . . . . . 7  |-  ( x  e.  RR  ->  ( -u -u x  e.  A  <->  x  e.  A ) )
106, 9bitrd 187 . . . . . 6  |-  ( x  e.  RR  ->  ( -u x  e.  { z  e.  RR  |  -u z  e.  A }  <->  x  e.  A ) )
1110biimprd 157 . . . . 5  |-  ( x  e.  RR  ->  (
x  e.  A  ->  -u x  e.  { z  e.  RR  |  -u z  e.  A }
) )
121, 11syli 37 . . . 4  |-  ( A 
C_  RR  ->  ( x  e.  A  ->  -u x  e.  { z  e.  RR  |  -u z  e.  A } ) )
13 elex2 2676 . . . 4  |-  ( -u x  e.  { z  e.  RR  |  -u z  e.  A }  ->  E. y 
y  e.  { z  e.  RR  |  -u z  e.  A }
)
1412, 13syl6 33 . . 3  |-  ( A 
C_  RR  ->  ( x  e.  A  ->  E. y 
y  e.  { z  e.  RR  |  -u z  e.  A }
) )
1514exlimdv 1775 . 2  |-  ( A 
C_  RR  ->  ( E. x  x  e.  A  ->  E. y  y  e. 
{ z  e.  RR  |  -u z  e.  A } ) )
1615imp 123 1  |-  ( ( A  C_  RR  /\  E. x  x  e.  A
)  ->  E. y 
y  e.  { z  e.  RR  |  -u z  e.  A }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1316   E.wex 1453    e. wcel 1465   {crab 2397    C_ wss 3041   RRcr 7587   -ucneg 7902
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-setind 4422  ax-resscn 7680  ax-1cn 7681  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-addcom 7688  ax-addass 7690  ax-distr 7692  ax-i2m1 7693  ax-0id 7696  ax-rnegex 7697  ax-cnre 7699
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-sub 7903  df-neg 7904
This theorem is referenced by: (None)
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