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Theorem negm 9407
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 3091 . . . . 5  |-  ( A 
C_  RR  ->  ( x  e.  A  ->  x  e.  RR ) )
2 renegcl 8023 . . . . . . . 8  |-  ( x  e.  RR  ->  -u x  e.  RR )
3 negeq 7955 . . . . . . . . . 10  |-  ( z  =  -u x  ->  -u z  =  -u -u x )
43eleq1d 2208 . . . . . . . . 9  |-  ( z  =  -u x  ->  ( -u z  e.  A  <->  -u -u x  e.  A ) )
54elrab3 2841 . . . . . . . 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 7753 . . . . . . . . 9  |-  ( x  e.  RR  ->  x  e.  CC )
87negnegd 8064 . . . . . . . 8  |-  ( x  e.  RR  ->  -u -u x  =  x )
98eleq1d 2208 . . . . . . 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 2702 . . . 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 1791 . 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 1331   E.wex 1468    e. wcel 1480   {crab 2420    C_ wss 3071   RRcr 7619   -ucneg 7934
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-setind 4452  ax-resscn 7712  ax-1cn 7713  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-addcom 7720  ax-addass 7722  ax-distr 7724  ax-i2m1 7725  ax-0id 7728  ax-rnegex 7729  ax-cnre 7731
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-sub 7935  df-neg 7936
This theorem is referenced by: (None)
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