Theorem tpid3gVD 16666

Description: Virtual deduction proof of tpid3g 3115.

Assertion

Ref	Expression
tpid3gVD	|- (A e. B -> A e. {C, D, A})

Proof of Theorem tpid3gVD

Step	Hyp	Ref	Expression
1		idn2 16509	. . . . . . 7 |- . A e. B, x = A   ⊢   x = A .
2		3mix3 1047	. . . . . . . . . 10 |- (x = A -> (x = C \/ x = D \/ x = A))
3	1, 2	e2 16521	. . . . . . . . 9 |- . A e. B, x = A   ⊢   (x = C \/ x = D \/ x = A) .
4		abid 1873	. . . . . . . . 9 |- (x e. {x | (x = C \/ x = D \/ x = A)} <-> (x = C \/ x = D \/ x = A))
5	3, 4	e2bir 16523	. . . . . . . 8 |- . A e. B, x = A   ⊢   x e. {x | (x = C \/ x = D \/ x = A)} .
6		dftp2 3075	. . . . . . . . 9 |- {C, D, A} = {x | (x = C \/ x = D \/ x = A)}
7	6	eleq2i 1961	. . . . . . . 8 |- (x e. {C, D, A} <-> x e. {x | (x = C \/ x = D \/ x = A)})
8	5, 7	e2bir 16523	. . . . . . 7 |- . A e. B, x = A   ⊢   x e. {C, D, A} .
9		eleq1 1957	. . . . . . . 8 |- (x = A -> (x e. {C, D, A} <-> A e. {C, D, A}))
10	9	biimpd 170	. . . . . . 7 |- (x = A -> (x e. {C, D, A} -> A e. {C, D, A}))
11	1, 8, 10	e22 16561	. . . . . 6 |- . A e. B, x = A   ⊢   A e. {C, D, A} .
12	11	in2 16506	. . . . 5 |- . A e. B   ⊢   (x = A -> A e. {C, D, A}) .
13	12	gen11 16511	. . . 4 |- . A e. B   ⊢   A.x(x = A -> A e. {C, D, A}) .
14		19.23v 1672	. . . 4 |- (A.x(x = A -> A e. {C, D, A}) <-> (E.x x = A -> A e. {C, D, A}))
15	13, 14	e1bi 16519	. . 3 |- . A e. B   ⊢   (E.x x = A -> A e. {C, D, A}) .
16		idn1 16484	. . . 4 |- . A e. B   ⊢   A e. B .
17		elex 2302	. . . 4 |- (A e. B -> E.x x = A)
18	16, 17	e1_ 16518	. . 3 |- . A e. B   ⊢   E.x x = A .
19		id 73	. . 3 |- ((E.x x = A -> A e. {C, D, A}) -> (E.x x = A -> A e. {C, D, A}))
20	15, 18, 19	e11 16578	. 2 |- . A e. B   ⊢   A e. {C, D, A} .
21	20	in1 16481	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 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  df-vd1 16480  df-vd2 16489

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