ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  negm Unicode version

Theorem negm 8770
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 2994 . . . . 5  |-  ( A 
C_  RR  ->  ( x  e.  A  ->  x  e.  RR ) )
2 renegcl 7425 . . . . . . . 8  |-  ( x  e.  RR  ->  -u x  e.  RR )
3 negeq 7357 . . . . . . . . . 10  |-  ( z  =  -u x  ->  -u z  =  -u -u x )
43eleq1d 2148 . . . . . . . . 9  |-  ( z  =  -u x  ->  ( -u z  e.  A  <->  -u -u x  e.  A ) )
54elrab3 2751 . . . . . . . 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 7157 . . . . . . . . 9  |-  ( x  e.  RR  ->  x  e.  CC )
87negnegd 7466 . . . . . . . 8  |-  ( x  e.  RR  ->  -u -u x  =  x )
98eleq1d 2148 . . . . . . 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 2616 . . . 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 1741 . 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 1285   E.wex 1422    e. wcel 1434   {crab 2353    C_ wss 2974   RRcr 7031   -ucneg 7336
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966  ax-setind 4282  ax-resscn 7119  ax-1cn 7120  ax-icn 7122  ax-addcl 7123  ax-addrcl 7124  ax-mulcl 7125  ax-addcom 7127  ax-addass 7129  ax-distr 7131  ax-i2m1 7132  ax-0id 7135  ax-rnegex 7136  ax-cnre 7138
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-br 3788  df-opab 3842  df-id 4050  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-iota 4891  df-fun 4928  df-fv 4934  df-riota 5493  df-ov 5540  df-oprab 5541  df-mpt2 5542  df-sub 7337  df-neg 7338
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator