Theorem tpid3gOLD 4306

Description: Obsolete proof of tpid3g 4305 as of 30-Apr-2021. Closed theorem form of tpid3 4307. This proof was automatically generated from the virtual deduction proof tpid3gVD 39077 using a translation program. (Contributed by Alan Sare, 24-Oct-2011.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
tpid3gOLD	|- ( A e. B -> A e. { C , D , A } )

Proof of Theorem tpid3gOLD

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elisset 3215	. 2 |- ( A e. B -> E. x x = A )
2		3mix3 1232	. . . . . . 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 2610	. . . . . 6 |- ( x e. { x | ( x = C \/ x = D \/ x = A ) } <-> ( x = C \/ x = D \/ x = A ) )
5	3, 4	syl6ibr 242	. . . . 5 |- ( A e. B -> ( x = A -> x e. { x | ( x = C \/ x = D \/ x = A ) } ) )
6		dftp2 4231	. . . . . 6 |- { C , D , A } = { x | ( x = C \/ x = D \/ x = A ) }
7	6	eleq2i 2693	. . . . 5 |- ( x e. { C , D , A } <-> x e. { x | ( x = C \/ x = D \/ x = A ) } )
8	5, 7	syl6ibr 242	. . . 4 |- ( A e. B -> ( x = A -> x e. { C , D , A } ) )
9		eleq1 2689	. . . 4 |- ( x = A -> ( x e. { C , D , A } <-> A e. { C , D , A } ) )
10	8, 9	mpbidi 231	. . 3 |- ( A e. B -> ( x = A -> A e. { C , D , A } ) )
11	10	exlimdv 1861	. 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 1036   = 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

This theorem is referenced by: (None)