Theorem tpid3g 3809

Description: Closed theorem form of tpid3 3810. (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 2830	. 2 |- ( A e. B -> E. x x = A )
2		3mix3 1195	. . . . . . 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 2222	. . . . . 6 |- ( x e. { x | ( x = C \/ x = D \/ x = A ) } <-> ( x = C \/ x = D \/ x = A ) )
5	3, 4	imbitrrdi 162	. . . . 5 |- ( A e. B -> ( x = A -> x e. { x | ( x = C \/ x = D \/ x = A ) } ) )
6		dftp2 3740	. . . . . 6 |- { C , D , A } = { x | ( x = C \/ x = D \/ x = A ) }
7	6	eleq2i 2301	. . . . 5 |- ( x e. { C , D , A } <-> x e. { x | ( x = C \/ x = D \/ x = A ) } )
8	5, 7	imbitrrdi 162	. . . 4 |- ( A e. B -> ( x = A -> x e. { C , D , A } ) )
9		eleq1 2297	. . . 4 |- ( x = A -> ( x e. { C , D , A } <-> A e. { C , D , A } ) )
10	8, 9	mpbidi 151	. . 3 |- ( A e. B -> ( x = A -> A e. { C , D , A } ) )
11	10	exlimdv 1868	. 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 1004   = wceq 1398   E.wex 1541   e. wcel 2205   {cab 2220   {ctp 3693

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3217  df-sn 3697  df-pr 3698  df-tp 3699

This theorem is referenced by:  rngmulrg  13368  srngmulrd  13379  lmodscad  13397  ipsmulrd  13409  ipsipd  13412  topgrptsetd  13429