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Theorem tpid3g 4085
Description: Closed theorem form of tpid3 4086. This proof was automatically generated from the virtual deduction proof tpid3gVD 31875 using a translation program. (Contributed by Alan Sare, 24-Oct-2011.)
Assertion
Ref Expression
tpid3g  |-  ( A  e.  B  ->  A  e.  { C ,  D ,  A } )

Proof of Theorem tpid3g
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elisset 3076 . 2  |-  ( A  e.  B  ->  E. x  x  =  A )
2 3mix3 1159 . . . . . . 7  |-  ( x  =  A  ->  (
x  =  C  \/  x  =  D  \/  x  =  A )
)
32a1i 11 . . . . . 6  |-  ( A  e.  B  ->  (
x  =  A  -> 
( x  =  C  \/  x  =  D  \/  x  =  A ) ) )
4 abid 2438 . . . . . 6  |-  ( x  e.  { x  |  ( x  =  C  \/  x  =  D  \/  x  =  A ) }  <->  ( x  =  C  \/  x  =  D  \/  x  =  A ) )
53, 4syl6ibr 227 . . . . 5  |-  ( A  e.  B  ->  (
x  =  A  ->  x  e.  { x  |  ( x  =  C  \/  x  =  D  \/  x  =  A ) } ) )
6 dftp2 4017 . . . . . 6  |-  { C ,  D ,  A }  =  { x  |  ( x  =  C  \/  x  =  D  \/  x  =  A ) }
76eleq2i 2527 . . . . 5  |-  ( x  e.  { C ,  D ,  A }  <->  x  e.  { x  |  ( x  =  C  \/  x  =  D  \/  x  =  A ) } )
85, 7syl6ibr 227 . . . 4  |-  ( A  e.  B  ->  (
x  =  A  ->  x  e.  { C ,  D ,  A }
) )
9 eleq1 2521 . . . 4  |-  ( x  =  A  ->  (
x  e.  { C ,  D ,  A }  <->  A  e.  { C ,  D ,  A }
) )
108, 9mpbidi 216 . . 3  |-  ( A  e.  B  ->  (
x  =  A  ->  A  e.  { C ,  D ,  A }
) )
1110exlimdv 1691 . 2  |-  ( A  e.  B  ->  ( E. x  x  =  A  ->  A  e.  { C ,  D ,  A } ) )
121, 11mpd 15 1  |-  ( A  e.  B  ->  A  e.  { C ,  D ,  A } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ w3o 964    = wceq 1370   E.wex 1587    e. wcel 1758   {cab 2436   {ctp 3976
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-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-v 3067  df-un 3428  df-sn 3973  df-pr 3975  df-tp 3977
This theorem is referenced by:  tpnzd  4092  f1dom3fv3dif  6076  f1dom3el3dif  6077  en3lplem1  7918  en3lp  7920  nb3graprlem1  23491  en3lplem1VD  31876  en3lpVD  31878
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