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Theorem tpid3g 3741
Description: Closed theorem form of tpid3 3742. This proof was automatically generated from the virtual deduction proof tpid3gVD 28618 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 2798 . 2  |-  ( A  e.  B  ->  E. x  x  =  A )
2 3mix3 1126 . . . . . . 7  |-  ( x  =  A  ->  (
x  =  C  \/  x  =  D  \/  x  =  A )
)
32a1i 10 . . . . . 6  |-  ( A  e.  B  ->  (
x  =  A  -> 
( x  =  C  \/  x  =  D  \/  x  =  A ) ) )
4 abid 2271 . . . . . 6  |-  ( x  e.  { x  |  ( x  =  C  \/  x  =  D  \/  x  =  A ) }  <->  ( x  =  C  \/  x  =  D  \/  x  =  A ) )
53, 4syl6ibr 218 . . . . 5  |-  ( A  e.  B  ->  (
x  =  A  ->  x  e.  { x  |  ( x  =  C  \/  x  =  D  \/  x  =  A ) } ) )
6 dftp2 3679 . . . . . 6  |-  { C ,  D ,  A }  =  { x  |  ( x  =  C  \/  x  =  D  \/  x  =  A ) }
76eleq2i 2347 . . . . 5  |-  ( x  e.  { C ,  D ,  A }  <->  x  e.  { x  |  ( x  =  C  \/  x  =  D  \/  x  =  A ) } )
85, 7syl6ibr 218 . . . 4  |-  ( A  e.  B  ->  (
x  =  A  ->  x  e.  { C ,  D ,  A }
) )
9 eleq1 2343 . . . 4  |-  ( x  =  A  ->  (
x  e.  { C ,  D ,  A }  <->  A  e.  { C ,  D ,  A }
) )
108, 9mpbidi 207 . . 3  |-  ( A  e.  B  ->  (
x  =  A  ->  A  e.  { C ,  D ,  A }
) )
1110exlimdv 1664 . 2  |-  ( A  e.  B  ->  ( E. x  x  =  A  ->  A  e.  { C ,  D ,  A } ) )
121, 11mpd 14 1  |-  ( A  e.  B  ->  A  e.  { C ,  D ,  A } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ w3o 933   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269   {ctp 3642
This theorem is referenced by:  en3lplem1  7416  en3lp  7418  en3lplem1VD  28619  en3lpVD  28621
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-un 3157  df-sn 3646  df-pr 3647  df-tp 3648
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