Theorem tpid3g 3831

Description: Closed theorem form of tpid3 3832. This proof was automatically generated from the virtual deduction proof tpid3gVD in set.mm 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 2869	. 2 |- (A e. B -> E.x x = A)
2		3mix3 1126	. . . . . . 7 |- (x = A -> (x = C \/ x = D \/ x = A))
3	2	a1i 10	. . . . . 6 |- (A e. B -> (x = A -> (x = C \/ x = D \/ x = A)))
4		abid 2341	. . . . . 6 |- (x e. {x | (x = C \/ x = D \/ x = A)} <-> (x = C \/ x = D \/ x = A))
5	3, 4	syl6ibr 218	. . . . 5 |- (A e. B -> (x = A -> x e. {x | (x = C \/ x = D \/ x = A)}))
6		dftp2 3772	. . . . . 6 |- {C, D, A} = {x | (x = C \/ x = D \/ x = A)}
7	6	eleq2i 2417	. . . . 5 |- (x e. {C, D, A} <-> x e. {x | (x = C \/ x = D \/ x = A)})
8	5, 7	syl6ibr 218	. . . 4 |- (A e. B -> (x = A -> x e. {C, D, A}))
9		eleq1 2413	. . . 4 |- (x = A -> (x e. {C, D, A} <-> A e. {C, D, A}))
10	8, 9	mpbidi 207	. . 3 |- (A e. B -> (x = A -> A e. {C, D, A}))
11	10	exlimdv 1636	. 2 |- (A e. B -> (E.x x = A -> A e. {C, D, A}))
12	1, 11	mpd 14	1 |- (A e. B -> A e. {C, D, A})

Syntax hints:   -> wi 4   \/ w3o 933  E.wex 1541   = wceq 1642   e. wcel 1710  {cab 2339  {ctp 3739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-un 3214  df-sn 3741  df-pr 3742  df-tp 3743

This theorem is referenced by: (None)