Theorem tpid3g 4112

Description: Closed theorem form of tpid3 4113. This proof was automatically generated from the virtual deduction proof tpid3gVD 37098 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 3092	. 2 |- ( A e. B -> E. x x = A )
2		3mix3 1176	. . . . . . 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 2409	. . . . . 6 |- ( x e. { x | ( x = C \/ x = D \/ x = A ) } <-> ( x = C \/ x = D \/ x = A ) )
5	3, 4	syl6ibr 230	. . . . 5 |- ( A e. B -> ( x = A -> x e. { x | ( x = C \/ x = D \/ x = A ) } ) )
6		dftp2 4043	. . . . . 6 |- { C , D , A } = { x | ( x = C \/ x = D \/ x = A ) }
7	6	eleq2i 2500	. . . . 5 |- ( x e. { C , D , A } <-> x e. { x | ( x = C \/ x = D \/ x = A ) } )
8	5, 7	syl6ibr 230	. . . 4 |- ( A e. B -> ( x = A -> x e. { C , D , A } ) )
9		eleq1 2494	. . . 4 |- ( x = A -> ( x e. { C , D , A } <-> A e. { C , D , A } ) )
10	8, 9	mpbidi 219	. . 3 |- ( A e. B -> ( x = A -> A e. { C , D , A } ) )
11	10	exlimdv 1768	. 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 981   = wceq 1437   E.wex 1659   e. wcel 1868   {cab 2407   {ctp 4000

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-v 3083  df-un 3441  df-sn 3997  df-pr 3999  df-tp 4001

This theorem is referenced by:  tpnzd  4119  f1dom3fv3dif  6179  f1dom3el3dif  6180  en3lplem1  8121  en3lp  8123  nb3graprlem1  25164  en3lplem1VD  37099  en3lpVD  37101  etransclem48OLD  37966  etransclem48  37967