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Theorem tpid3g 3638
Description: Closed theorem form of tpid3 3639. (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 2700 . 2  |-  ( A  e.  B  ->  E. x  x  =  A )
2 3mix3 1152 . . . . . . 7  |-  ( x  =  A  ->  (
x  =  C  \/  x  =  D  \/  x  =  A )
)
32a1i 9 . . . . . 6  |-  ( A  e.  B  ->  (
x  =  A  -> 
( x  =  C  \/  x  =  D  \/  x  =  A ) ) )
4 abid 2127 . . . . . 6  |-  ( x  e.  { x  |  ( x  =  C  \/  x  =  D  \/  x  =  A ) }  <->  ( x  =  C  \/  x  =  D  \/  x  =  A ) )
53, 4syl6ibr 161 . . . . 5  |-  ( A  e.  B  ->  (
x  =  A  ->  x  e.  { x  |  ( x  =  C  \/  x  =  D  \/  x  =  A ) } ) )
6 dftp2 3572 . . . . . 6  |-  { C ,  D ,  A }  =  { x  |  ( x  =  C  \/  x  =  D  \/  x  =  A ) }
76eleq2i 2206 . . . . 5  |-  ( x  e.  { C ,  D ,  A }  <->  x  e.  { x  |  ( x  =  C  \/  x  =  D  \/  x  =  A ) } )
85, 7syl6ibr 161 . . . 4  |-  ( A  e.  B  ->  (
x  =  A  ->  x  e.  { C ,  D ,  A }
) )
9 eleq1 2202 . . . 4  |-  ( x  =  A  ->  (
x  e.  { C ,  D ,  A }  <->  A  e.  { C ,  D ,  A }
) )
108, 9mpbidi 150 . . 3  |-  ( A  e.  B  ->  (
x  =  A  ->  A  e.  { C ,  D ,  A }
) )
1110exlimdv 1791 . 2  |-  ( A  e.  B  ->  ( E. x  x  =  A  ->  A  e.  { C ,  D ,  A } ) )
121, 11mpd 13 1  |-  ( A  e.  B  ->  A  e.  { C ,  D ,  A } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ w3o 961    = wceq 1331   E.wex 1468    e. wcel 1480   {cab 2125   {ctp 3529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3or 963  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-tp 3535
This theorem is referenced by:  rngmulrg  12091  srngmulrd  12098  lmodscad  12109  ipsmulrd  12117  ipsipd  12120  topgrptsetd  12127
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