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Theorem tpid3g 3921
Description: Closed theorem form of tpid3 3922. This proof was automatically generated from the virtual deduction proof tpid3gVD 29016 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 2968 . 2  |-  ( A  e.  B  ->  E. x  x  =  A )
2 3mix3 1129 . . . . . . 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 2426 . . . . . 6  |-  ( x  e.  { x  |  ( x  =  C  \/  x  =  D  \/  x  =  A ) }  <->  ( x  =  C  \/  x  =  D  \/  x  =  A ) )
53, 4syl6ibr 220 . . . . 5  |-  ( A  e.  B  ->  (
x  =  A  ->  x  e.  { x  |  ( x  =  C  \/  x  =  D  \/  x  =  A ) } ) )
6 dftp2 3856 . . . . . 6  |-  { C ,  D ,  A }  =  { x  |  ( x  =  C  \/  x  =  D  \/  x  =  A ) }
76eleq2i 2502 . . . . 5  |-  ( x  e.  { C ,  D ,  A }  <->  x  e.  { x  |  ( x  =  C  \/  x  =  D  \/  x  =  A ) } )
85, 7syl6ibr 220 . . . 4  |-  ( A  e.  B  ->  (
x  =  A  ->  x  e.  { C ,  D ,  A }
) )
9 eleq1 2498 . . . 4  |-  ( x  =  A  ->  (
x  e.  { C ,  D ,  A }  <->  A  e.  { C ,  D ,  A }
) )
108, 9mpbidi 209 . . 3  |-  ( A  e.  B  ->  (
x  =  A  ->  A  e.  { C ,  D ,  A }
) )
1110exlimdv 1647 . 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 set class
Syntax hints:    -> wi 4    \/ w3o 936   E.wex 1551    = wceq 1653    e. wcel 1726   {cab 2424   {ctp 3818
This theorem is referenced by:  en3lplem1  7672  en3lp  7674  nb3graprlem1  21462  en3lplem1VD  29017  en3lpVD  29019
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960  df-un 3327  df-sn 3822  df-pr 3823  df-tp 3824
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