Theorem tpid3g 4086

Description: Closed theorem form of tpid3 4087. This proof was automatically generated from the virtual deduction proof tpid3gVD 37232 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.

Step	Hyp	Ref	Expression
1		elisset 3056	. 2 |- ( A e. B -> E. x x = A )
2		3mix3 1178	. . . . . . 7 |- ( x = A -> ( x = C \/ x = D \/ x = A ) )
3	2	a1i 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 ) )
5	3, 4	syl6ibr 231	. . . . 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 ) }
7	6	eleq2i 2520	. . . . 5 |- ( x e. { C , D , A } <-> x e. { x | ( x = C \/ x = D \/ x = A ) } )
8	5, 7	syl6ibr 231	. . . 4 |- ( A e. B -> ( x = A -> x e. { C , D , A } ) )
9		eleq1 2516	. . . 4 |- ( x = A -> ( x e. { C , D , A } <-> A e. { C , D , A } ) )
10	8, 9	mpbidi 220	. . 3 |- ( A e. B -> ( x = A -> A e. { C , D , A } ) )
11	10	exlimdv 1778	. 2 |- ( A e. B -> ( E. x x = A -> A e. { C , D , A } ) )
12	1, 11	mpd 15	1 |- ( A e. B -> A e. { C , D , A } )

Syntax hints:   -> wi 4   \/ w3o 983   = wceq 1443   E.wex 1662   e. wcel 1886   {cab 2436   {ctp 3971

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-v 3046  df-un 3408  df-sn 3968  df-pr 3970  df-tp 3972

This theorem is referenced by:  tpnzd  4093  f1dom3fv3dif  6166  f1dom3el3dif  6167  en3lplem1  8116  en3lp  8118  nb3graprlem1  25172  en3lplem1VD  37233  en3lpVD  37235  etransclem48OLD  38141  etransclem48  38142  nb3grprlem1  39437  cplgr3v  39485