Theorem tpid3g 3115

Description: Closed theorem form of tpid3 3116. This proof was automatically generated from the virtual deduction proof tpid3gVD 16666 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

Step	Hyp	Ref	Expression
1		elex 2302	. 2 |- (A e. B -> E.x x = A)
2		3mix3 1047	. . . . . . . 8 |- (x = A -> (x = C \/ x = D \/ x = A))
3	2	a1i 8	. . . . . . 7 |- (A e. B -> (x = A -> (x = C \/ x = D \/ x = A)))
4		abid 1873	. . . . . . 7 |- (x e. {x | (x = C \/ x = D \/ x = A)} <-> (x = C \/ x = D \/ x = A))
5	3, 4	syl6ibr 230	. . . . . 6 |- (A e. B -> (x = A -> x e. {x | (x = C \/ x = D \/ x = A)}))
6		dftp2 3075	. . . . . . 7 |- {C, D, A} = {x | (x = C \/ x = D \/ x = A)}
7	6	eleq2i 1961	. . . . . 6 |- (x e. {C, D, A} <-> x e. {x | (x = C \/ x = D \/ x = A)})
8	5, 7	syl6ibr 230	. . . . 5 |- (A e. B -> (x = A -> x e. {C, D, A}))
9		eleq1 1957	. . . . 5 |- (x = A -> (x e. {C, D, A} <-> A e. {C, D, A}))
10	8, 9	mpbidi 649	. . . 4 |- (A e. B -> (x = A -> A e. {C, D, A}))
11	10	19.21aiv 1664	. . 3 |- (A e. B -> A.x(x = A -> A e. {C, D, A}))
12		19.23v 1672	. . 3 |- (A.x(x = A -> A e. {C, D, A}) <-> (E.x x = A -> A e. {C, D, A}))
13	11, 12	sylib 215	. 2 |- (A e. B -> (E.x x = A -> A e. {C, D, A}))
14	1, 13	mpd 29	1 |- (A e. B -> A e. {C, D, A})

Syntax hints:   -> wi 3   \/ w3o 857  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326  {cab 1871  {ctp 3051

This theorem is referenced by:  en3lplem1 5756  en3lp 5758  en3lplem1VD 16667  en3lpVD 16669

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-un 2600  df-sn 3049  df-pr 3050  df-tp 3052

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