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Theorem negm 9069
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 3017 . . . . 5  |-  ( A 
C_  RR  ->  ( x  e.  A  ->  x  e.  RR ) )
2 renegcl 7722 . . . . . . . 8  |-  ( x  e.  RR  ->  -u x  e.  RR )
3 negeq 7654 . . . . . . . . . 10  |-  ( z  =  -u x  ->  -u z  =  -u -u x )
43eleq1d 2156 . . . . . . . . 9  |-  ( z  =  -u x  ->  ( -u z  e.  A  <->  -u -u x  e.  A ) )
54elrab3 2770 . . . . . . . 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 7454 . . . . . . . . 9  |-  ( x  e.  RR  ->  x  e.  CC )
87negnegd 7763 . . . . . . . 8  |-  ( x  e.  RR  ->  -u -u x  =  x )
98eleq1d 2156 . . . . . . 7  |-  ( x  e.  RR  ->  ( -u -u x  e.  A  <->  x  e.  A ) )
106, 9bitrd 186 . . . . . 6  |-  ( x  e.  RR  ->  ( -u x  e.  { z  e.  RR  |  -u z  e.  A }  <->  x  e.  A ) )
1110biimprd 156 . . . . 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 2635 . . . 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 1747 . 2  |-  ( A 
C_  RR  ->  ( E. x  x  e.  A  ->  E. y  y  e. 
{ z  e.  RR  |  -u z  e.  A } ) )
1615imp 122 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 102    <-> wb 103    = wceq 1289   E.wex 1426    e. wcel 1438   {crab 2363    C_ wss 2997   RRcr 7328   -ucneg 7633
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-setind 4343  ax-resscn 7416  ax-1cn 7417  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-addcom 7424  ax-addass 7426  ax-distr 7428  ax-i2m1 7429  ax-0id 7432  ax-rnegex 7433  ax-cnre 7435
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-iota 4967  df-fun 5004  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-sub 7634  df-neg 7635
This theorem is referenced by: (None)
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