Theorem tpid3g 3646

Description: Closed theorem form of tpid3 3647. (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 2703	. 2 |- ( A e. B -> E. x x = A )
2		3mix3 1153	. . . . . . 7 |- ( x = A -> ( x = C \/ x = D \/ x = A ) )
3	2	a1i 9	. . . . . 6 |- ( A e. B -> ( x = A -> ( x = C \/ x = D \/ x = A ) ) )
4		abid 2128	. . . . . 6 |- ( x e. { x | ( x = C \/ x = D \/ x = A ) } <-> ( x = C \/ x = D \/ x = A ) )
5	3, 4	syl6ibr 161	. . . . 5 |- ( A e. B -> ( x = A -> x e. { x | ( x = C \/ x = D \/ x = A ) } ) )
6		dftp2 3580	. . . . . 6 |- { C , D , A } = { x | ( x = C \/ x = D \/ x = A ) }
7	6	eleq2i 2207	. . . . 5 |- ( x e. { C , D , A } <-> x e. { x | ( x = C \/ x = D \/ x = A ) } )
8	5, 7	syl6ibr 161	. . . 4 |- ( A e. B -> ( x = A -> x e. { C , D , A } ) )
9		eleq1 2203	. . . 4 |- ( x = A -> ( x e. { C , D , A } <-> A e. { C , D , A } ) )
10	8, 9	mpbidi 150	. . 3 |- ( A e. B -> ( x = A -> A e. { C , D , A } ) )
11	10	exlimdv 1792	. 2 |- ( A e. B -> ( E. x x = A -> A e. { C , D , A } ) )
12	1, 11	mpd 13	1 |- ( A e. B -> A e. { C , D , A } )

Syntax hints:   -> wi 4   \/ w3o 962   = wceq 1332   E.wex 1469   e. wcel 1481   {cab 2126   {ctp 3534

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3or 964  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3080  df-sn 3538  df-pr 3539  df-tp 3540

This theorem is referenced by:  rngmulrg  12116  srngmulrd  12123  lmodscad  12134  ipsmulrd  12142  ipsipd  12145  topgrptsetd  12152