Theorem tpid3gVD 39077

Description: Virtual deduction proof of tpid3g 4305. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
tpid3gVD	|- ( A e. B -> A e. { C , D , A } )

Proof of Theorem tpid3gVD

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		idn2 38838	. . . . . . 7 |- (. A e. B ,. x = A ->. x = A ).
2		3mix3 1232	. . . . . . . . . 10 |- ( x = A -> ( x = C \/ x = D \/ x = A ) )
3	1, 2	e2 38856	. . . . . . . . 9 |- (. A e. B ,. x = A ->. ( x = C \/ x = D \/ x = A ) ).
4		abid 2610	. . . . . . . . 9 |- ( x e. { x | ( x = C \/ x = D \/ x = A ) } <-> ( x = C \/ x = D \/ x = A ) )
5	3, 4	e2bir 38858	. . . . . . . 8 |- (. A e. B ,. x = A ->. x e. { x | ( x = C \/ x = D \/ x = A ) } ).
6		dftp2 4231	. . . . . . . . 9 |- { C , D , A } = { x | ( x = C \/ x = D \/ x = A ) }
7	6	eleq2i 2693	. . . . . . . 8 |- ( x e. { C , D , A } <-> x e. { x | ( x = C \/ x = D \/ x = A ) } )
8	5, 7	e2bir 38858	. . . . . . 7 |- (. A e. B ,. x = A ->. x e. { C , D , A } ).
9		eleq1 2689	. . . . . . . 8 |- ( x = A -> ( x e. { C , D , A } <-> A e. { C , D , A } ) )
10	9	biimpd 219	. . . . . . 7 |- ( x = A -> ( x e. { C , D , A } -> A e. { C , D , A } ) )
11	1, 8, 10	e22 38896	. . . . . 6 |- (. A e. B ,. x = A ->. A e. { C , D , A } ).
12	11	in2 38830	. . . . 5 |- (. A e. B ->. ( x = A -> A e. { C , D , A } ) ).
13	12	gen11 38841	. . . 4 |- (. A e. B ->. A. x ( x = A -> A e. { C , D , A } ) ).
14		19.23v 1902	. . . 4 |- ( A. x ( x = A -> A e. { C , D , A } ) <-> ( E. x x = A -> A e. { C , D , A } ) )
15	13, 14	e1bi 38854	. . 3 |- (. A e. B ->. ( E. x x = A -> A e. { C , D , A } ) ).
16		idn1 38790	. . . 4 |- (. A e. B ->. A e. B ).
17		elisset 3215	. . . 4 |- ( A e. B -> E. x x = A )
18	16, 17	e1a 38852	. . 3 |- (. A e. B ->. E. x x = A ).
19		id 22	. . 3 |- ( ( E. x x = A -> A e. { C , D , A } ) -> ( E. x x = A -> A e. { C , D , A } ) )
20	15, 18, 19	e11 38913	. 2 |- (. A e. B ->. A e. { C , D , A } ).
21	20	in1 38787	1 |- ( A e. B -> A e. { C , D , A } )

Syntax hints:   -> wi 4   \/ w3o 1036   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   {ctp 4181

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-un 3579  df-sn 4178  df-pr 4180  df-tp 4182  df-vd1 38786  df-vd2 38794

This theorem is referenced by: (None)