Theorem trint 3426

Description: The intersection of a class of transitive sets is transitive. Exercise 5(b) of [Enderton] p. 73. (Contributed by Scott Fenton, 25-Feb-2011.)

Assertion

Ref	Expression
trint	|- (A.x e. A Tr x -> Tr |^|A)

Distinct variable group:   x,A

Proof of Theorem trint

Step	Hyp	Ref	Expression
1		dftr3 3415	. . . . . 6 |- (Tr x <-> A.y e. x y C_ x)
2	1	ralbii 2127	. . . . 5 |- (A.x e. A Tr x <-> A.x e. A A.y e. x y C_ x)
3	2	biimpi 168	. . . 4 |- (A.x e. A Tr x -> A.x e. A A.y e. x y C_ x)
4		df-ral 2109	. . . . . 6 |- (A.y e. x y C_ x <-> A.y(y e. x -> y C_ x))
5	4	ralbii 2127	. . . . 5 |- (A.x e. A A.y e. x y C_ x <-> A.x e. A A.y(y e. x -> y C_ x))
6		ralcom4 2310	. . . . 5 |- (A.x e. A A.y(y e. x -> y C_ x) <-> A.yA.x e. A (y e. x -> y C_ x))
7	5, 6	bitri 190	. . . 4 |- (A.x e. A A.y e. x y C_ x <-> A.yA.x e. A (y e. x -> y C_ x))
8	3, 7	sylib 215	. . 3 |- (A.x e. A Tr x -> A.yA.x e. A (y e. x -> y C_ x))
9		ralim 2164	. . . 4 |- (A.x e. A (y e. x -> y C_ x) -> (A.x e. A y e. x -> A.x e. A y C_ x))
10	9	alimi 1338	. . 3 |- (A.yA.x e. A (y e. x -> y C_ x) -> A.y(A.x e. A y e. x -> A.x e. A y C_ x))
11	8, 10	syl 12	. 2 |- (A.x e. A Tr x -> A.y(A.x e. A y e. x -> A.x e. A y C_ x))
12		dftr3 3415	. . 3 |- (Tr |^|A <-> A.y e. |^|Ay C_ |^|A)
13		df-ral 2109	. . 3 |- (A.y e. |^|Ay C_ |^|A <-> A.y(y e. |^|A -> y C_ |^|A))
14		visset 2295	. . . . . 6 |- y e. _V
15	14	elint2 3221	. . . . 5 |- (y e. |^|A <-> A.x e. A y e. x)
16		ssint 3232	. . . . 5 |- (y C_ |^|A <-> A.x e. A y C_ x)
17	15, 16	imbi12i 205	. . . 4 |- ((y e. |^|A -> y C_ |^|A) <-> (A.x e. A y e. x -> A.x e. A y C_ x))
18	17	albii 1346	. . 3 |- (A.y(y e. |^|A -> y C_ |^|A) <-> A.y(A.x e. A y e. x -> A.x e. A y C_ x))
19	12, 13, 18	3bitri 194	. 2 |- (Tr |^|A <-> A.y(A.x e. A y e. x -> A.x e. A y C_ x))
20	11, 19	sylibr 217	1 |- (A.x e. A Tr x -> Tr |^|A)

Syntax hints:   -> wi 3  A.wal 1296   e. wcel 1300  A.wral 2105   C_ wss 2593  |^|cint 3214  Tr wtr 3411

This theorem is referenced by:  dfon2lem8 13856

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-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ral 2109  df-v 2294  df-in 2603  df-ss 2605  df-uni 3178  df-int 3215  df-tr 3412

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